(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 72879, 2259] NotebookOptionsPosition[ 66663, 2048] NotebookOutlinePosition[ 67002, 2063] CellTagsIndexPosition[ 66959, 2060] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"FusionAtlasPaths", "=", RowBox[{"{", RowBox[{ "\"\<~/projects/fusionatlas/code/package/\>\"", ",", "\"\<~/Documents/math papers/fusionatlas/code/package/\>\"", ",", "\"\<~/Documents/Math/fusionatlas/code/package/\>\"", ",", "\"\<~/Documents/fusionatlas/code/package\>\""}], "}"}]}], ";"}], "\n", RowBox[{ RowBox[{"$Path", "=", RowBox[{"$Path", "~", "Join", "~", "FusionAtlasPaths"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"<<", "FusionAtlas`"}]}], "Input", CellChangeTimes->{{3.462819492529354*^9, 3.4628195014751987`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Loading FusionAtlas` version 0\\n\"\>", "\[InvisibleSpace]", "\<\"Read more at \ http://tqft.net/wiki/Atlas_of_subfactors\"\>"}], SequenceForm[ "Loading FusionAtlas` version 0\n", "Read more at http://tqft.net/wiki/Atlas_of_subfactors"], Editable->False]], "Print", CellChangeTimes->{3.468533254581893*^9, 3.468711033168055*^9, 3.4687212004757233`*^9, 3.468779935449456*^9, 3.469456085029009*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Found precomputed data in \"\>", "\[InvisibleSpace]", \ "\<\"/Users/scott/projects/fusionatlas/code/data\"\>"}], SequenceForm[ "Found precomputed data in ", "/Users/scott/projects/fusionatlas/code/data"], Editable->False]], "Print", CellChangeTimes->{3.468533254581893*^9, 3.468711033168055*^9, 3.4687212004757233`*^9, 3.468779935449456*^9, 3.4694560850327463`*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"DisplayBigraph", "[", RowBox[{"\[CapitalGamma]", "=", "HaagerupBigraph"}], "]"}]], "Input", CellChangeTimes->{{3.468533371889179*^9, 3.468533381605977*^9}}], Cell[BoxData[ GraphicsBox[{{PointBox[{1, 1}], PointBox[{2, 1}], PointBox[{3, 1}], PointBox[{4, 1}], {PointBox[{5, 0}], PointBox[{5, 2}]}, {PointBox[{6, 0}], PointBox[{6, 2}]}, {PointBox[{7, 0}], PointBox[{7, 2}]}}, { LineBox[{{1, 1}, {2, 1}}], LineBox[{{2, 1}, {3, 1}}], LineBox[{{3, 1}, {4, 1}}], {LineBox[{{4, 1}, {5, 0}}], LineBox[{{4, 1}, {5, 2}}]}, {{LineBox[{{5, 0}, {6, 0}}], {}}, {{}, LineBox[{{5, 2}, {6, 2}}]}}, {{LineBox[{{6, 0}, {7, 0}}], {}}, {{}, LineBox[{{6, 2}, {7, 2}}]}}}, {{}, {}, {}, {{}, {}}, {{{}, {}}, {{}, {}}}, {{{}, {}}, {{}, \ {}}}}}]], "Output", CellChangeTimes->{3.468533382009717*^9, 3.468533429459146*^9, 3.468711037417395*^9, 3.468721204688833*^9, 3.468779965616302*^9, 3.469456108022107*^9}] }, Open ]], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"P", ",", "A", ",", "Z", ",", "B", ",", "Bp"}], "]"}]], "Input", CellChangeTimes->{{3.468533426766057*^9, 3.468533427826735*^9}, { 3.468722870907257*^9, 3.468722876770981*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]_GradedBigraph", "]"}], "[", "n_Integer", "]"}], "/;", RowBox[{"n", "==", RowBox[{"GraphRank", "[", "\[CapitalGamma]", "]"}]}]}], ":=", RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "n", "]"}], "=", RowBox[{"Function", "[", RowBox[{"x", ",", RowBox[{"Evaluate", "[", RowBox[{"CharacteristicPolynomial", "[", RowBox[{ RowBox[{"GraphAdjacencyMatrix", "[", "\[CapitalGamma]", "]"}], ",", "x"}], "]"}], "]"}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.468533266585536*^9, 3.46853334830828*^9}, { 3.468533404198649*^9, 3.468533406605674*^9}, {3.4687257498991337`*^9, 3.468725767200342*^9}, {3.468779603493277*^9, 3.468779604077186*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]_GradedBigraph", "]"}], "[", "n_Integer", "]"}], "/;", RowBox[{"n", "\[Equal]", RowBox[{ RowBox[{"GraphRank", "[", "\[CapitalGamma]", "]"}], "+", "1"}]}]}], ":=", RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "n", "]"}], "=", RowBox[{"Function", "[", RowBox[{"x", ",", RowBox[{"Evaluate", "[", RowBox[{"-", RowBox[{"CharacteristicPolynomial", "[", RowBox[{ RowBox[{"GraphAdjacencyMatrix", "[", RowBox[{"Translate", "[", RowBox[{"\[CapitalGamma]", ",", "1"}], "]"}], "]"}], ",", "x"}], "]"}]}], "]"}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.468533266585536*^9, 3.468533369120837*^9}, { 3.46853342144689*^9, 3.4685334242686033`*^9}, 3.468722866878264*^9, { 3.4687257716881*^9, 3.468725783023555*^9}, {3.468779606325485*^9, 3.4687796069577312`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]_GradedBigraph", "]"}], "[", "n_Integer", "]"}], "/;", RowBox[{"n", "\[GreaterEqual]", RowBox[{ RowBox[{"GraphRank", "[", "\[CapitalGamma]", "]"}], "+", "2"}]}]}], ":=", RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "n", "]"}], "=", RowBox[{"Function", "[", RowBox[{"x", ",", RowBox[{"Evaluate", "[", RowBox[{"Expand", "[", RowBox[{ RowBox[{"x", " ", RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", RowBox[{"n", "-", "1"}], "]"}], "[", "x", "]"}]}], "-", RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", RowBox[{"n", "-", "2"}], "]"}], "[", "x", "]"}]}], "]"}], "]"}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.4685334434628773`*^9, 3.468533519703555*^9}, { 3.468725788498111*^9, 3.468725789729595*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]_GradedBigraph", "]"}], "[", "n_Integer", "]"}], "/;", RowBox[{"n", "<", RowBox[{"GraphRank", "[", "\[CapitalGamma]", "]"}]}]}], ":=", RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "n", "]"}], "=", RowBox[{"Function", "[", RowBox[{"x", ",", RowBox[{"Evaluate", "[", RowBox[{"Expand", "[", RowBox[{ RowBox[{"x", " ", RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", RowBox[{"n", "+", "1"}], "]"}], "[", "x", "]"}]}], "-", RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", RowBox[{"n", "+", "2"}], "]"}], "[", "x", "]"}]}], "]"}], "]"}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.468722117467626*^9, 3.468722168219619*^9}, 3.46872579338451*^9, {3.469456122418803*^9, 3.4694561240801*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"PP", "[", "\[CapitalGamma]_GradedBigraph", "]"}], "[", RowBox[{ RowBox[{"x_", "?", "NumericQ"}], ",", "n_Integer"}], "]"}], "/;", RowBox[{ RowBox[{"(", RowBox[{"n", "\[Equal]", RowBox[{"GraphRank", "[", "\[CapitalGamma]", "]"}]}], ")"}], "\[Or]", RowBox[{"(", RowBox[{"n", "==", RowBox[{ RowBox[{"GraphRank", "[", "\[CapitalGamma]", "]"}], "+", "1"}]}], ")"}]}]}], ":=", RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "n", "]"}], "[", "x", "]"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"PP", "[", "\[CapitalGamma]_GradedBigraph", "]"}], "[", RowBox[{ RowBox[{"x_", "?", "NumericQ"}], ",", "n_Integer"}], "]"}], "/;", RowBox[{"(", RowBox[{"n", "\[GreaterEqual]", RowBox[{ RowBox[{"GraphRank", "[", "\[CapitalGamma]", "]"}], "+", "2"}]}], ")"}]}], ":=", RowBox[{ RowBox[{ RowBox[{"PP", "[", "\[CapitalGamma]", "]"}], "[", RowBox[{"x", ",", "n"}], "]"}], "=", RowBox[{ RowBox[{"x", " ", RowBox[{ RowBox[{"PP", "[", "\[CapitalGamma]", "]"}], "[", RowBox[{"x", ",", RowBox[{"n", "-", "1"}]}], "]"}]}], "-", RowBox[{ RowBox[{"PP", "[", "\[CapitalGamma]", "]"}], "[", RowBox[{"x", ",", RowBox[{"n", "-", "2"}]}], "]"}]}]}]}]}], "Input", CellChangeTimes->{{3.468779498736171*^9, 3.468779668595265*^9}, { 3.468779779888658*^9, 3.468779824238908*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"P", "[", RowBox[{"Drop", "[", RowBox[{"\[CapitalGamma]", ",", "2"}], "]"}], "]"}], "[", "8", "]"}], "[", "x", "]"}]], "Input", CellChangeTimes->{{3.468722348179674*^9, 3.468722359788026*^9}, 3.468725808941413*^9}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"-", "8"}], " ", SuperscriptBox["x", "2"]}], "+", RowBox[{"14", " ", SuperscriptBox["x", "4"]}], "-", RowBox[{"7", " ", SuperscriptBox["x", "6"]}], "+", SuperscriptBox["x", "8"]}]], "Output", CellChangeTimes->{{3.468722355608614*^9, 3.4687223602556334`*^9}, 3.468722880119895*^9, {3.46872579888281*^9, 3.4687258095554647`*^9}, { 3.469456092057794*^9, 3.469456126619831*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "10", "]"}], "[", "x", "]"}]], "Input", CellChangeTimes->{{3.468722361459077*^9, 3.468722363811392*^9}, 3.468725814125043*^9}], Cell[BoxData[ RowBox[{ RowBox[{"12", " ", SuperscriptBox["x", "2"]}], "-", RowBox[{"32", " ", SuperscriptBox["x", "4"]}], "+", RowBox[{"27", " ", SuperscriptBox["x", "6"]}], "-", RowBox[{"9", " ", SuperscriptBox["x", "8"]}], "+", SuperscriptBox["x", "10"]}]], "Output", CellChangeTimes->{ 3.468722364187619*^9, 3.468722880817068*^9, 3.468725814751198*^9, { 3.469456092272839*^9, 3.469456127525689*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "11", "]"}], "[", "x", "]"}]], "Input", CellChangeTimes->{3.4687228386303167`*^9, 3.4687258990394087`*^9}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"-", "4"}], " ", "x"}], "+", RowBox[{"30", " ", SuperscriptBox["x", "3"]}], "-", RowBox[{"52", " ", SuperscriptBox["x", "5"]}], "+", RowBox[{"35", " ", SuperscriptBox["x", "7"]}], "-", RowBox[{"10", " ", SuperscriptBox["x", "9"]}], "+", SuperscriptBox["x", "11"]}]], "Output", CellChangeTimes->{3.468722839302895*^9, 3.4687228815136967`*^9, 3.468725900086935*^9, 3.4694561305538197`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"P", "[", RowBox[{"Translate", "[", RowBox[{"\[CapitalGamma]", ",", "2"}], "]"}], "]"}], "[", "12", "]"}], "[", "x", "]"}], "-", RowBox[{"Expand", "[", RowBox[{"x", " ", RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "11", "]"}], "[", "x", "]"}]}], "]"}], "+", RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "10", "]"}], "[", "x", "]"}]}]], "Input", CellChangeTimes->{ 3.468722437016588*^9, {3.468722695032564*^9, 3.468722701278551*^9}, { 3.468722748830626*^9, 3.4687227751935387`*^9}, {3.4687259043750896`*^9, 3.468725941196015*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{ 3.4687224377710876`*^9, {3.468722696235793*^9, 3.4687227016471033`*^9}, { 3.468722750264749*^9, 3.468722776192883*^9}, 3.468722883140684*^9, { 3.46872590073459*^9, 3.468725904667983*^9}, 3.4687259420489283`*^9, 3.469456144694229*^9}] }, Open ]], Cell[BoxData[ RowBox[{"Clear", "[", "A", "]"}]], "Input", CellChangeTimes->{{3.468533982250662*^9, 3.468533983093011*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]_GradedBigraph", "]"}], ":=", RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "=", RowBox[{"Function", "[", RowBox[{"t", ",", RowBox[{"Evaluate", "[", RowBox[{"Factor", "[", RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{"n", "=", RowBox[{"GraphRank", "[", "\[CapitalGamma]", "]"}]}], "}"}], ",", RowBox[{ FractionBox[ RowBox[{"t", "-", SuperscriptBox["t", RowBox[{"-", "1"}]]}], RowBox[{ SuperscriptBox["t", RowBox[{"n", "+", "2"}]], "-", SuperscriptBox["t", "n"]}]], RowBox[{"(", RowBox[{ RowBox[{"t", " ", RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", RowBox[{"n", "+", "1"}], "]"}], "[", RowBox[{"t", "+", SuperscriptBox["t", RowBox[{"-", "1"}]]}], "]"}]}], "-", RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "n", "]"}], "[", RowBox[{"t", "+", SuperscriptBox["t", RowBox[{"-", "1"}]]}], "]"}]}], ")"}]}]}], "]"}], "]"}], "]"}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.4685337411834497`*^9, 3.4685338089954853`*^9}, { 3.468533850312429*^9, 3.468533980441341*^9}, {3.468710952991115*^9, 3.4687109546213284`*^9}, {3.468725958295925*^9, 3.4687259610521297`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}]], "Input", CellChangeTimes->{ 3.4687222265027113`*^9, {3.468779958734528*^9, 3.468779959886587*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", SuperscriptBox["t", "2"]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", SuperscriptBox["t", "4"]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "-", SuperscriptBox["t", "2"], "-", SuperscriptBox["t", "4"], "+", SuperscriptBox["t", "6"]}], ")"}]}], SuperscriptBox["t", "11"]]], "Output", CellChangeTimes->{ 3.468722226880755*^9, 3.468722894312457*^9, {3.468725955231621*^9, 3.468725967813252*^9}, {3.468779960143517*^9, 3.468779967366169*^9}, 3.469456149186129*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Expand", "[", RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "12", "]"}], "[", RowBox[{"t", "+", SuperscriptBox["t", RowBox[{"-", "1"}]]}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.4687821688104687`*^9, 3.468782197715495*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"-", "8"}], "+", FractionBox["1", SuperscriptBox["t", "12"]], "+", FractionBox["1", SuperscriptBox["t", "10"]], "-", FractionBox["2", SuperscriptBox["t", "6"]], "-", FractionBox["5", SuperscriptBox["t", "4"]], "-", FractionBox["7", SuperscriptBox["t", "2"]], "-", RowBox[{"7", " ", SuperscriptBox["t", "2"]}], "-", RowBox[{"5", " ", SuperscriptBox["t", "4"]}], "-", RowBox[{"2", " ", SuperscriptBox["t", "6"]}], "+", SuperscriptBox["t", "10"], "+", SuperscriptBox["t", "12"]}]], "Output", CellChangeTimes->{{3.468782179248592*^9, 3.468782198120976*^9}, 3.4694561502346478`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{"n", "=", "5"}], "}"}], ",", RowBox[{"Simplify", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "n", "]"}], "[", RowBox[{"t", "+", SuperscriptBox["t", RowBox[{"-", "1"}]]}], "]"}], RowBox[{"(", RowBox[{"t", "-", SuperscriptBox["t", RowBox[{"-", "1"}]]}], ")"}]}], "-", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["t", "n"], RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}]}], "-", RowBox[{ SuperscriptBox["t", RowBox[{"-", "n"}]], RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", SuperscriptBox["t", RowBox[{"-", "1"}]], "]"}]}]}], ")"}]}], "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.468539410402989*^9, 3.468539482211636*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{{3.468539472170649*^9, 3.4685394826267433`*^9}, 3.468721204776627*^9, 3.468722924872295*^9, 3.468725990651618*^9, 3.468779968263936*^9, 3.469456151497727*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"CyclotomicFactors", "[", RowBox[{"a_", ",", "t_"}], "]"}], ":=", RowBox[{"Times", "@@", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"#", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], RowBox[{"#", "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}]], "&"}], "/@", RowBox[{"Cases", "[", RowBox[{ RowBox[{"FactorList", "[", "a", "]"}], ",", RowBox[{ RowBox[{"{", RowBox[{"p_", ",", "n_"}], "}"}], "/;", RowBox[{"p", "\[Equal]", RowBox[{"Cyclotomic", "[", RowBox[{ RowBox[{"2", RowBox[{"Exponent", "[", RowBox[{"p", ",", "t"}], "]"}]}], ",", "t"}], "]"}]}]}]}], "]"}]}], ")"}]}]}]], "Input", CellChangeTimes->{{3.4687109609514437`*^9, 3.4687110166488533`*^9}, { 3.468711057180025*^9, 3.4687110764831123`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"CyclotomicFactors", "[", RowBox[{ RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}], ",", "t"}], "]"}]], "Input", CellChangeTimes->{{3.468711042469686*^9, 3.468711047951892*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", SuperscriptBox["t", "2"]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", SuperscriptBox["t", "4"]}], ")"}]}]], "Output", CellChangeTimes->{{3.46871104862948*^9, 3.468711077363574*^9}, 3.468721204825891*^9, 3.468722928456861*^9, 3.468725996812194*^9, 3.4687799694957533`*^9, 3.469456154709156*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"Z", "[", "\[CapitalGamma]_GradedBigraph", "]"}], ":=", RowBox[{ RowBox[{"Z", "[", "\[CapitalGamma]", "]"}], "=", RowBox[{"Function", "[", RowBox[{"t", ",", RowBox[{"Evaluate", "[", RowBox[{"CyclotomicFactors", "[", RowBox[{ RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}], ",", "t"}], "]"}], "]"}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.468711127169077*^9, 3.4687111342464437`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"B", "[", "\[CapitalGamma]_GradedBigraph", "]"}], ":=", RowBox[{ RowBox[{"B", "[", "\[CapitalGamma]", "]"}], "=", RowBox[{"Function", "[", RowBox[{"t", ",", RowBox[{"Evaluate", "[", RowBox[{"Factor", "[", RowBox[{ RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}], "/", RowBox[{ RowBox[{"Z", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}]}], "]"}], "]"}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.468711079236786*^9, 3.468711142566572*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"B", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}]], "Input", CellChangeTimes->{{3.468711147863471*^9, 3.468711151043561*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"-", "1"}], "-", SuperscriptBox["t", "2"], "-", SuperscriptBox["t", "4"], "+", SuperscriptBox["t", "6"]}], SuperscriptBox["t", "11"]]], "Output", CellChangeTimes->{3.468711151399933*^9, 3.468721204882071*^9, 3.468722932044258*^9, 3.4687260004447403`*^9, 3.468779970869294*^9, 3.469456157687866*^9}] }, Open ]], Cell[BoxData[{ RowBox[{"Clear", "[", "Bp", "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Bp", "[", "\[CapitalGamma]_GradedBigraph", "]"}], ":=", RowBox[{ RowBox[{"Bp", "[", "\[CapitalGamma]", "]"}], "=", RowBox[{"Function", "[", RowBox[{"t", ",", RowBox[{"Evaluate", "[", RowBox[{"Factor", "[", RowBox[{"D", "[", RowBox[{ RowBox[{ RowBox[{"B", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}], ",", "t"}], "]"}], "]"}], "]"}]}], "]"}]}]}]}], "Input", CellChangeTimes->{{3.468711186605576*^9, 3.468711189921194*^9}, { 3.46871132527404*^9, 3.468711361504422*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Bp", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}]], "Input", CellChangeTimes->{{3.468711346216185*^9, 3.468711350791613*^9}}], Cell[BoxData[ RowBox[{"-", FractionBox[ RowBox[{ RowBox[{"-", "11"}], "-", RowBox[{"9", " ", SuperscriptBox["t", "2"]}], "-", RowBox[{"7", " ", SuperscriptBox["t", "4"]}], "+", RowBox[{"5", " ", SuperscriptBox["t", "6"]}]}], SuperscriptBox["t", "12"]]}]], "Output", CellChangeTimes->{{3.4687113479123373`*^9, 3.468711362486909*^9}, 3.468721204931375*^9, 3.468722934647645*^9, 3.4687260033015327`*^9, 3.468779971892963*^9, 3.4694561593350697`*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"minBp", "[", "\[CapitalGamma]_", "]"}], ":=", RowBox[{"Sqrt", "[", RowBox[{ RowBox[{"Minimize", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"ExpToTrig", "[", RowBox[{"Expand", "[", RowBox[{ RowBox[{ RowBox[{"Bp", "[", "\[CapitalGamma]", "]"}], "[", RowBox[{"Exp", "[", RowBox[{"2", "\[Pi]", " ", "\[ImaginaryI]", " ", "\[Theta]"}], "]"}], "]"}], RowBox[{ RowBox[{"Bp", "[", "\[CapitalGamma]", "]"}], "[", RowBox[{"Exp", "[", RowBox[{ RowBox[{"-", "2"}], "\[Pi]", " ", "\[ImaginaryI]", " ", "\[Theta]"}], "]"}], "]"}]}], "]"}], "]"}], ",", RowBox[{"0", "\[LessEqual]", "\[Theta]", "<", "1"}]}], "}"}], ",", RowBox[{"{", "\[Theta]", "}"}]}], "]"}], "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], "]"}]}]], "Input", CellChangeTimes->{{3.4687113674192877`*^9, 3.468711432660345*^9}, { 3.468711475379449*^9, 3.468711478849525*^9}, {3.4687115492294207`*^9, 3.468711550397243*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", RowBox[{"minBp", "[", "\[CapitalGamma]", "]"}], "]"}]], "Input", CellChangeTimes->{{3.46871143396532*^9, 3.468711435977838*^9}, { 3.468711484792348*^9, 3.468711494328211*^9}}], Cell[BoxData["1.4724419747988307`"], "Output", CellChangeTimes->{3.468711495270934*^9, 3.468711551741363*^9, 3.468721205638467*^9, 3.468722940414721*^9, 3.468726010887405*^9, 3.468779983125681*^9, 3.469456162347719*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"maxBp", "[", "\[CapitalGamma]_", "]"}], ":=", RowBox[{"Sqrt", "[", RowBox[{"-", RowBox[{ RowBox[{"Minimize", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", RowBox[{"ExpToTrig", "[", RowBox[{"Expand", "[", RowBox[{ RowBox[{ RowBox[{"Bp", "[", "\[CapitalGamma]", "]"}], "[", RowBox[{"Exp", "[", RowBox[{"2", "\[Pi]", " ", "\[ImaginaryI]", " ", "\[Theta]"}], "]"}], "]"}], RowBox[{ RowBox[{"Bp", "[", "\[CapitalGamma]", "]"}], "[", RowBox[{"Exp", "[", RowBox[{ RowBox[{"-", "2"}], "\[Pi]", " ", "\[ImaginaryI]", " ", "\[Theta]"}], "]"}], "]"}]}], "]"}], "]"}]}], ",", RowBox[{"0", "\[LessEqual]", "\[Theta]", "<", "1"}]}], "}"}], ",", RowBox[{"{", "\[Theta]", "}"}]}], "]"}], "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.4687113674192877`*^9, 3.468711432660345*^9}, { 3.468711475379449*^9, 3.4687115610925913`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", RowBox[{"maxBp", "[", "\[CapitalGamma]", "]"}], "]"}]], "Input", CellChangeTimes->{{3.46871143396532*^9, 3.468711435977838*^9}, { 3.468711484792348*^9, 3.4687115174223633`*^9}}], Cell[BoxData["23.166728733287115`"], "Output", CellChangeTimes->{{3.468711495270934*^9, 3.468711519120461*^9}, 3.4687115626450777`*^9, 3.4687212060816383`*^9, 3.468722944253207*^9, 3.468726013842689*^9, 3.468779984300686*^9, 3.469456163634067*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"RepeatedRootBound", "[", "\[CapitalGamma]_", "]"}], ":=", RowBox[{"Ceiling", "[", FractionBox[ RowBox[{"maxBp", "[", "\[CapitalGamma]", "]"}], RowBox[{"2", RowBox[{"minBp", "[", "\[CapitalGamma]", "]"}]}]], "]"}]}]], "Input", CellChangeTimes->{{3.468711568234717*^9, 3.4687115968089657`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"RepeatedRootBound", "[", "\[CapitalGamma]", "]"}]], "Input", CellChangeTimes->{{3.4687115979412622`*^9, 3.468711602023012*^9}}], Cell[BoxData["8"], "Output", CellChangeTimes->{3.4687116029353228`*^9, 3.468721206641334*^9, 3.468722947826682*^9, 3.4687260199523993`*^9, 3.4687799863939533`*^9, 3.469456166610578*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"length", "[", RowBox[{"a_", ",", "t_"}], "]"}], ":=", RowBox[{"Length", "[", RowBox[{"Exponent", "[", RowBox[{"a", ",", "t", ",", "List"}], "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.468721255101425*^9, 3.468721266721187*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"length", "[", RowBox[{ RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}], ",", "t"}], "]"}]], "Input", CellChangeTimes->{{3.4687212681647663`*^9, 3.468721275287408*^9}}], Cell[BoxData["6"], "Output", CellChangeTimes->{3.468721275802944*^9, 3.468726030521077*^9, 3.4687799866462297`*^9, 3.469456167252939*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"PrimesUpTo", "[", "k_Integer", "]"}], "/;", RowBox[{"k", "\[GreaterEqual]", "2"}]}], ":=", RowBox[{ RowBox[{"Reap", "[", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"m", "=", "1"}], "}"}], ",", RowBox[{"While", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"m", "=", RowBox[{"NextPrime", "[", "m", "]"}]}], ")"}], "\[LessEqual]", "k"}], ",", RowBox[{"Sow", "[", "m", "]"}]}], "]"}]}], "]"}], "]"}], "\[LeftDoubleBracket]", RowBox[{"2", ",", "1"}], "\[RightDoubleBracket]"}]}]], "Input", CellChangeTimes->{{3.468721294374807*^9, 3.46872133869525*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"PrimesUpTo", "[", "12", "]"}]], "Input", CellChangeTimes->{{3.468721340802236*^9, 3.4687213432867737`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"2", ",", "3", ",", "5", ",", "7", ",", "11"}], "}"}]], "Output", CellChangeTimes->{3.468721343749267*^9, 3.4687799881377287`*^9, 3.46945616891628*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"RootOfUnityBound", "[", "\[CapitalGamma]_", "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "t", "}"}], ",", RowBox[{"2", RowBox[{"(", RowBox[{"Times", "@@", RowBox[{"PrimesUpTo", "[", RowBox[{"2", RowBox[{"length", "[", RowBox[{ RowBox[{ RowBox[{"B", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}], ",", "t"}], "]"}]}], "]"}]}], ")"}], RowBox[{"(", RowBox[{ RowBox[{"Exponent", "[", RowBox[{ RowBox[{ RowBox[{"B", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}], ",", "t", ",", "Max"}], "]"}], "-", RowBox[{"Exponent", "[", RowBox[{ RowBox[{ RowBox[{"B", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}], ",", "t", ",", "Min"}], "]"}]}], ")"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.468721285776619*^9, 3.4687212929636097`*^9}, { 3.468721347160263*^9, 3.4687214160605288`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"RootOfUnityBound", "[", "\[CapitalGamma]", "]"}]], "Input", CellChangeTimes->{{3.468721392193471*^9, 3.468721400466155*^9}}], Cell[BoxData["2520"], "Output", CellChangeTimes->{{3.468721400859845*^9, 3.46872141737593*^9}, 3.468722954499077*^9, 3.468726041977304*^9, 3.468779989259412*^9, 3.469456170513838*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"CheckRoot", "[", RowBox[{"\[CapitalGamma]_", ",", "k_"}], "]"}], ":=", RowBox[{ RowBox[{"CheckRoot", "[", RowBox[{"\[CapitalGamma]", ",", "k"}], "]"}], "=", RowBox[{"(", RowBox[{ RowBox[{"$MaxRootDegree", "=", RowBox[{"Max", "[", RowBox[{"$MaxRootDegree", ",", RowBox[{"k", "+", "10"}]}], "]"}]}], ";", RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{"\[Zeta]", "=", RowBox[{"ToNumberField", "[", RowBox[{"Exp", "[", RowBox[{"2", "\[Pi]", " ", RowBox[{"\[ImaginaryI]", "/", "k"}]}], "]"}], "]"}]}], "}"}], ",", RowBox[{"Or", "@@", RowBox[{"Flatten", "[", RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{"CoprimeQ", "[", RowBox[{"z", ",", "k"}], "]"}], "\[And]", RowBox[{ RowBox[{ RowBox[{"PP", "[", "\[CapitalGamma]", "]"}], "[", RowBox[{ RowBox[{ SuperscriptBox["\[Zeta]", "z"], "+", SuperscriptBox["\[Zeta]", RowBox[{"-", "z"}]]}], ",", "n"}], "]"}], "\[Equal]", "0"}]}], ",", RowBox[{"{", RowBox[{"z", ",", "1", ",", "1"}], RowBox[{"(*", RowBox[{"0", ",", RowBox[{"k", "-", "1"}]}], "*)"}], "}"}], ",", RowBox[{"{", RowBox[{"n", ",", RowBox[{"GraphRank", "[", "\[CapitalGamma]", "]"}], ",", RowBox[{ RowBox[{"GraphRank", "[", "\[CapitalGamma]", "]"}], "+", "k", "-", "1"}]}], "}"}]}], "]"}], "]"}]}]}], "]"}]}], ")"}]}]}]], "Input", CellChangeTimes->{{3.468723971621738*^9, 3.468724050743671*^9}, { 3.468724091175597*^9, 3.468724115460414*^9}, {3.468724260548023*^9, 3.468724334398569*^9}, {3.468724428861188*^9, 3.4687244494974737`*^9}, { 3.468726068952868*^9, 3.468726100901699*^9}, {3.468726940496163*^9, 3.4687270424303217`*^9}, {3.468730587958493*^9, 3.46873060901934*^9}, { 3.468766862352851*^9, 3.46876687931106*^9}, {3.468766912781596*^9, 3.4687669161102037`*^9}, 3.4687669633230143`*^9, {3.468779999130981*^9, 3.46878001384864*^9}, {3.468780065215032*^9, 3.46878007286138*^9}, { 3.468780362797637*^9, 3.468780366171604*^9}, {3.468780501903295*^9, 3.468780513718498*^9}, {3.469624541818446*^9, 3.469624563672345*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"CheckRoot", "[", RowBox[{"\[CapitalGamma]", ",", "8"}], "]"}]], "Input", CellChangeTimes->{{3.468724023743126*^9, 3.468724034567544*^9}, { 3.468724281215796*^9, 3.4687242813273983`*^9}}], Cell[BoxData["True"], "Output", CellChangeTimes->{{3.468724028957882*^9, 3.468724051664469*^9}, { 3.468724110640655*^9, 3.468724116816058*^9}, {3.468724278756941*^9, 3.468724335909587*^9}, 3.468724451189848*^9, {3.468726083108203*^9, 3.4687261018811693`*^9}, {3.468727027875223*^9, 3.468727045237637*^9}, { 3.468766888951618*^9, 3.46876691914421*^9}, 3.468780019029812*^9, 3.4687800763758087`*^9, 3.469456184919074*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"CheckRoot", "[", RowBox[{"\[CapitalGamma]", ",", "1050"}], "]"}]], "Input", CellChangeTimes->{{3.469624594049912*^9, 3.469624598089919*^9}}], Cell[BoxData["False"], "Output", CellChangeTimes->{3.469624611709059*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Dynamic", "[", "nn", "]"}]], "Input", CellChangeTimes->{{3.469456197860692*^9, 3.469456199785812*^9}}], Cell[BoxData[ DynamicBox[ToBoxes[$CellContext`nn, StandardForm], ImageSizeCache->{29., {1., 8.}}]], "Output", CellChangeTimes->{3.469456200239706*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Cases", "[", RowBox[{ RowBox[{"Range", "[", "100", "]"}], ",", RowBox[{"k_", "/;", RowBox[{"(", RowBox[{ RowBox[{"nn", "=", "k"}], ";", RowBox[{"CheckRoot", "[", RowBox[{"\[CapitalGamma]", ",", "k"}], "]"}]}], ")"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.4687242232967997`*^9, 3.468724246578527*^9}, { 3.468724402146665*^9, 3.468724402967245*^9}, {3.468726113034206*^9, 3.4687261349664583`*^9}, {3.4687669689099092`*^9, 3.468766974315099*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"3", ",", "4", ",", "6", ",", "8", ",", "16"}], "}"}]], "Output", CellChangeTimes->{{3.468726113476516*^9, 3.4687261308981123`*^9}, 3.468726221761407*^9, 3.468727043366186*^9, 3.468729639974934*^9, 3.4687306401138897`*^9, 3.468766906517549*^9, 3.468766964351143*^9, 3.468769762058495*^9, 3.468780286661578*^9, 3.468780375746222*^9, { 3.4687804995681257`*^9, 3.468780522001232*^9}, 3.4694561932515497`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Cases", "[", RowBox[{ RowBox[{"Range", "[", "2530", "]"}], ",", RowBox[{"k_", "/;", RowBox[{"(", RowBox[{ RowBox[{"nn", "=", "k"}], ";", RowBox[{"CheckRoot", "[", RowBox[{"\[CapitalGamma]", ",", "k"}], "]"}]}], ")"}]}]}], "]"}]], "Input", CellChangeTimes->{3.46878053136339*^9}], Cell[BoxData["$Aborted"], "Output", CellChangeTimes->{3.469808226598472*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"SumOfSquaresConstant", "[", "\[CapitalGamma]_", "]"}], ":=", RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{"n0", "=", RowBox[{"Min", "[", RowBox[{"0", ",", RowBox[{ RowBox[{"-", "2"}], RowBox[{"Exponent", "[", RowBox[{ RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}], ",", "t", ",", "Min"}], "]"}]}]}], "]"}]}], "}"}], ",", RowBox[{ RowBox[{"Plus", "@@", RowBox[{"(", SuperscriptBox[ RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "x", "}"}], ",", RowBox[{"x", "/.", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "n0", "]"}], "[", "x", "]"}], "\[Equal]", "0"}], ",", "x"}], "]"}]}]}], "]"}], "2"], ")"}]}], "-", RowBox[{"2", "n0"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.468725292093696*^9, 3.468725297764914*^9}, { 3.468725524768258*^9, 3.4687256209790173`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SumOfSquaresConstant", "[", "\[CapitalGamma]", "]"}]], "Input", CellChangeTimes->{{3.468725622215796*^9, 3.4687256292788763`*^9}}], Cell[BoxData["18"], "Output", CellChangeTimes->{3.468725629671021*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Expand", "[", RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "20", "]"}], "[", RowBox[{"t", "+", SuperscriptBox["t", RowBox[{"-", "1"}]]}], "]"}], "]"}]], "Input", CellChangeTimes->{3.4685398252257137`*^9}], Cell[BoxData[ RowBox[{"40", "-", FractionBox["1", SuperscriptBox["t", "30"]], "-", FractionBox["3", SuperscriptBox["t", "28"]], "-", FractionBox["4", SuperscriptBox["t", "26"]], "-", FractionBox["2", SuperscriptBox["t", "24"]], "+", FractionBox["5", SuperscriptBox["t", "22"]], "+", FractionBox["17", SuperscriptBox["t", "20"]], "+", FractionBox["30", SuperscriptBox["t", "18"]], "+", FractionBox["40", SuperscriptBox["t", "16"]], "+", FractionBox["44", SuperscriptBox["t", "14"]], "+", FractionBox["44", SuperscriptBox["t", "12"]], "+", FractionBox["42", SuperscriptBox["t", "10"]], "+", FractionBox["40", SuperscriptBox["t", "8"]], "+", FractionBox["40", SuperscriptBox["t", "6"]], "+", FractionBox["40", SuperscriptBox["t", "4"]], "+", FractionBox["40", SuperscriptBox["t", "2"]], "+", RowBox[{"40", " ", SuperscriptBox["t", "2"]}], "+", RowBox[{"40", " ", SuperscriptBox["t", "4"]}], "+", RowBox[{"40", " ", SuperscriptBox["t", "6"]}], "+", RowBox[{"40", " ", SuperscriptBox["t", "8"]}], "+", RowBox[{"42", " ", SuperscriptBox["t", "10"]}], "+", RowBox[{"44", " ", SuperscriptBox["t", "12"]}], "+", RowBox[{"44", " ", SuperscriptBox["t", "14"]}], "+", RowBox[{"40", " ", SuperscriptBox["t", "16"]}], "+", RowBox[{"30", " ", SuperscriptBox["t", "18"]}], "+", RowBox[{"17", " ", SuperscriptBox["t", "20"]}], "+", RowBox[{"5", " ", SuperscriptBox["t", "22"]}], "-", RowBox[{"2", " ", SuperscriptBox["t", "24"]}], "-", RowBox[{"4", " ", SuperscriptBox["t", "26"]}], "-", RowBox[{"3", " ", SuperscriptBox["t", "28"]}], "-", SuperscriptBox["t", "30"]}]], "Output", CellChangeTimes->{3.4685398258131123`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Expand", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "20", "]"}], "[", RowBox[{"t", "+", SuperscriptBox["t", RowBox[{"-", "1"}]]}], "]"}], RowBox[{"(", RowBox[{"t", "-", SuperscriptBox["t", RowBox[{"-", "1"}]]}], ")"}]}], "]"}]], "Input", CellChangeTimes->{{3.468539734975462*^9, 3.4685397539522743`*^9}}], Cell[BoxData[ RowBox[{ FractionBox["1", SuperscriptBox["t", "31"]], "+", FractionBox["2", SuperscriptBox["t", "29"]], "+", FractionBox["1", SuperscriptBox["t", "27"]], "-", FractionBox["2", SuperscriptBox["t", "25"]], "-", FractionBox["7", SuperscriptBox["t", "23"]], "-", FractionBox["12", SuperscriptBox["t", "21"]], "-", FractionBox["13", SuperscriptBox["t", "19"]], "-", FractionBox["10", SuperscriptBox["t", "17"]], "-", FractionBox["4", SuperscriptBox["t", "15"]], "+", FractionBox["2", SuperscriptBox["t", "11"]], "+", FractionBox["2", SuperscriptBox["t", "9"]], "-", RowBox[{"2", " ", SuperscriptBox["t", "9"]}], "-", RowBox[{"2", " ", SuperscriptBox["t", "11"]}], "+", RowBox[{"4", " ", SuperscriptBox["t", "15"]}], "+", RowBox[{"10", " ", SuperscriptBox["t", "17"]}], "+", RowBox[{"13", " ", SuperscriptBox["t", "19"]}], "+", RowBox[{"12", " ", SuperscriptBox["t", "21"]}], "+", RowBox[{"7", " ", SuperscriptBox["t", "23"]}], "+", RowBox[{"2", " ", SuperscriptBox["t", "25"]}], "-", SuperscriptBox["t", "27"], "-", RowBox[{"2", " ", SuperscriptBox["t", "29"]}], "-", SuperscriptBox["t", "31"]}]], "Output", CellChangeTimes->{{3.468539735796835*^9, 3.468539757277874*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}]], "Input", CellChangeTimes->{{3.4685339388456907`*^9, 3.468533987948997*^9}, { 3.46853932623545*^9, 3.468539361878821*^9}, {3.468539517271041*^9, 3.46853951906036*^9}, 3.468539691187376*^9, {3.468539724976954*^9, 3.4685397251741743`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox["2", SuperscriptBox["t", "11"]]}], "-", FractionBox["2", SuperscriptBox["t", "9"]], "+", FractionBox["4", SuperscriptBox["t", "5"]], "+", FractionBox["10", SuperscriptBox["t", "3"]], "+", FractionBox["13", "t"], "+", RowBox[{"12", " ", "t"}], "+", RowBox[{"7", " ", SuperscriptBox["t", "3"]}], "+", RowBox[{"2", " ", SuperscriptBox["t", "5"]}], "-", SuperscriptBox["t", "7"], "-", RowBox[{"2", " ", SuperscriptBox["t", "9"]}], "-", SuperscriptBox["t", "11"]}]], "Output", CellChangeTimes->{{3.468533940110734*^9, 3.4685339884624043`*^9}, { 3.468539329318839*^9, 3.46853936269985*^9}, 3.468539519361517*^9, 3.468539691676054*^9, 3.468539726152117*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", "%", "]"}]], "Input", CellChangeTimes->{{3.468539693717229*^9, 3.4685396953768597`*^9}}], Cell[BoxData[ RowBox[{"7", "-", FractionBox["2", SuperscriptBox["t", "10"]], "+", FractionBox["4", SuperscriptBox["t", "4"]], "+", FractionBox["6", SuperscriptBox["t", "2"]], "+", RowBox[{"5", " ", SuperscriptBox["t", "2"]}], "+", RowBox[{"2", " ", SuperscriptBox["t", "4"]}], "-", SuperscriptBox["t", "8"], "-", SuperscriptBox["t", "10"]}]], "Output", CellChangeTimes->{3.468539695759487*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Sort", "[", RowBox[{"N", "[", RowBox[{"Abs", "[", RowBox[{"x", "/.", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "10", "]"}], "[", "x", "]"}], "\[Equal]", "0"}], ",", "x"}], "]"}]}], "]"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.468534254277309*^9, 3.468534260505911*^9}, { 3.468534292336173*^9, 3.468534339853551*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ "0.`", ",", "0.`", ",", "0.45306504511165596`", ",", "0.45306504511165596`", ",", "0.45306504511165596`", ",", "0.45306504511165596`", ",", "1.`", ",", "1.`", ",", "1.0374904511506384`", ",", "1.0374904511506384`", ",", "1.4142135623730951`", ",", "1.4142135623730951`", ",", "1.5442838529140217`", ",", "1.5442838529140217`", ",", "1.5442838529140217`", ",", "1.5442838529140217`", ",", "1.8926214654226583`", ",", "1.8926214654226583`", ",", "2.0806869894950495`", ",", "2.0806869894950495`"}], "}"}]], "Output", CellChangeTimes->{{3.4685343059976273`*^9, 3.468534340265278*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Expand", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "50", "]"}], "[", RowBox[{"t", "+", SuperscriptBox["t", RowBox[{"-", "1"}]]}], "]"}], RowBox[{"(", RowBox[{"t", "-", SuperscriptBox["t", RowBox[{"-", "1"}]]}], ")"}]}], "]"}]], "Input", CellChangeTimes->{{3.46853338364154*^9, 3.4685334309466467`*^9}, { 3.468533511775098*^9, 3.468533579411249*^9}}], Cell[BoxData[ RowBox[{ FractionBox["1", SuperscriptBox["t", "61"]], "+", FractionBox["2", SuperscriptBox["t", "59"]], "+", FractionBox["1", SuperscriptBox["t", "57"]], "-", FractionBox["2", SuperscriptBox["t", "55"]], "-", FractionBox["7", SuperscriptBox["t", "53"]], "-", FractionBox["12", SuperscriptBox["t", "51"]], "-", FractionBox["13", SuperscriptBox["t", "49"]], "-", FractionBox["10", SuperscriptBox["t", "47"]], "-", FractionBox["4", SuperscriptBox["t", "45"]], "+", FractionBox["2", SuperscriptBox["t", "41"]], "+", FractionBox["2", SuperscriptBox["t", "39"]], "-", RowBox[{"2", " ", SuperscriptBox["t", "39"]}], "-", RowBox[{"2", " ", SuperscriptBox["t", "41"]}], "+", RowBox[{"4", " ", SuperscriptBox["t", "45"]}], "+", RowBox[{"10", " ", SuperscriptBox["t", "47"]}], "+", RowBox[{"13", " ", SuperscriptBox["t", "49"]}], "+", RowBox[{"12", " ", SuperscriptBox["t", "51"]}], "+", RowBox[{"7", " ", SuperscriptBox["t", "53"]}], "+", RowBox[{"2", " ", SuperscriptBox["t", "55"]}], "-", SuperscriptBox["t", "57"], "-", RowBox[{"2", " ", SuperscriptBox["t", "59"]}], "-", SuperscriptBox["t", "61"]}]], "Output", CellChangeTimes->{{3.4685333864009237`*^9, 3.468533431281888*^9}, { 3.468533512684572*^9, 3.4685335802259197`*^9}}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"GaloisGroupGrowthConstant", "[", "\[CapitalGamma]_", "]"}], ":=", FractionBox[ RowBox[{"Log", "[", "2", "]"}], RowBox[{"HomomorphismConstant", "[", "\[CapitalGamma]", "]"}]]}]], "Input",\ CellChangeTimes->{{3.4685361386477957`*^9, 3.468536150722105*^9}, { 3.468536313991371*^9, 3.468536322167595*^9}, 3.468536716164804*^9, { 3.468539063315798*^9, 3.468539064226502*^9}, 3.46853922340924*^9, 3.468539656757306*^9}], Cell[BoxData[ RowBox[{ RowBox[{"HomomorphismConstant", "[", "\[CapitalGamma]_GradedBigraph", "]"}], ":=", RowBox[{"HomomorphismConstant", "[", RowBox[{"Function", "[", RowBox[{"t", ",", RowBox[{"Evaluate", "[", FractionBox[ RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", SuperscriptBox["t", RowBox[{"-", "1"}]], "]"}], RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}]], "]"}]}], "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.468536215615913*^9, 3.468536279314619*^9}, { 3.468553011276154*^9, 3.468553033805146*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox["2", SuperscriptBox["t", "11"]]}], "-", FractionBox["2", SuperscriptBox["t", "9"]], "+", FractionBox["4", SuperscriptBox["t", "5"]], "+", FractionBox["10", SuperscriptBox["t", "3"]], "+", FractionBox["13", "t"], "+", RowBox[{"12", " ", "t"}], "+", RowBox[{"7", " ", SuperscriptBox["t", "3"]}], "+", RowBox[{"2", " ", SuperscriptBox["t", "5"]}], "-", SuperscriptBox["t", "7"], "-", RowBox[{"2", " ", SuperscriptBox["t", "9"]}], "-", SuperscriptBox["t", "11"]}]], "Output", CellChangeTimes->{3.468540088990587*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"4", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"Exponent", "[", RowBox[{ RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}], ",", "t", ",", "Max"}], "]"}], "-", RowBox[{"Exponent", "[", RowBox[{ RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}], ",", "t", ",", "Min"}], "]"}]}], ")"}], "/", "2"}], "+", "1", "-", "2"}], ")"}]}]], "Input", CellChangeTimes->{{3.468540028021316*^9, 3.468540215087685*^9}}], Cell[BoxData["10"], "Output", CellChangeTimes->{{3.468540042928331*^9, 3.4685401613287783`*^9}, 3.468540196585765*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"RootsGreaterThan2Bound", "[", "\[CapitalGamma]_", "]"}], ":=", RowBox[{"2", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"Exponent", "[", RowBox[{ RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}], ",", "t", ",", "Max"}], "]"}], "-", RowBox[{"Exponent", "[", RowBox[{ RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}], ",", "t", ",", "Min"}], "]"}]}], ")"}], "/", "2"}], "+", "1", "-", "2"}], ")"}]}]}]], "Input", CellChangeTimes->{{3.4685363987706842`*^9, 3.46853640415275*^9}, { 3.468536468957171*^9, 3.4685364748044147`*^9}, {3.468539253367153*^9, 3.468539253805634*^9}, 3.46854002488455*^9, {3.468540208344474*^9, 3.468540219169875*^9}, 3.4685410895366087`*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"RootsGreaterThan2Bound", "[", "\[CapitalGamma]", "]"}]], "Input", CellChangeTimes->{{3.4685363987706842`*^9, 3.46853640415275*^9}, { 3.468536468957171*^9, 3.4685364748044147`*^9}, {3.468539253367153*^9, 3.468539253805634*^9}, 3.46854002488455*^9, {3.468540208344474*^9, 3.468540219169875*^9}, 3.4685410895366087`*^9, 3.4685450480988493`*^9}], Cell[BoxData["40"], "Output", CellChangeTimes->{3.4685450484923487`*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"RootsGreaterThan2Bound", "[", "\[CapitalGamma]_", "]"}], ":=", RowBox[{"GraphRank", "[", "\[CapitalGamma]", "]"}]}]], "Input", CellChangeTimes->{{3.4685363987706842`*^9, 3.46853640415275*^9}, { 3.468536468957171*^9, 3.4685364748044147`*^9}, {3.468539253367153*^9, 3.468539253805634*^9}, 3.46854002488455*^9, {3.468540208344474*^9, 3.468540219169875*^9}, 3.4685410895366087`*^9, {3.468545059153326*^9, 3.468545061875108*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"RootsGreaterThan2Bound", "[", "\[CapitalGamma]", "]"}]], "Input", CellChangeTimes->{{3.4685363987706842`*^9, 3.46853640415275*^9}, { 3.468536468957171*^9, 3.4685364748044147`*^9}, {3.468539253367153*^9, 3.468539253805634*^9}, 3.46854002488455*^9, {3.468540208344474*^9, 3.468540219169875*^9}, 3.4685410895366087`*^9, 3.4685450480988493`*^9}], Cell[BoxData["10"], "Output", CellChangeTimes->{3.468545068278905*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"HomomorphismConstant", "[", "f_Function", "]"}], ":="}]], "Input", CellChangeTimes->{{3.468536294744549*^9, 3.468536304082079*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["\[Rho]", "\[Infinity]"], "[", "\[CapitalGamma]_GradedBigraph", "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "t", "}"}], ",", RowBox[{"Max", "[", RowBox[{"Cases", "[", RowBox[{ RowBox[{"t", "/.", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}], "\[Equal]", "0"}], ",", "t"}], "]"}]}], ",", RowBox[{"c_", "/;", RowBox[{ RowBox[{"Im", "[", "c", "]"}], "\[Equal]", "0"}]}]}], "]"}], "]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.468536974795369*^9, 3.4685369868897*^9}, { 3.4685370407719812`*^9, 3.468537091019886*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ SubscriptBox["\[Rho]", "\[Infinity]"], "[", "\[CapitalGamma]", "]"}]], "Input", CellChangeTimes->{{3.468537057965788*^9, 3.468537063028359*^9}}], Cell[BoxData[ SqrtBox[ RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"2", "-", RowBox[{"2", " ", SuperscriptBox["#1", "2"]}], "-", RowBox[{"4", " ", SuperscriptBox["#1", "3"]}], "-", RowBox[{"4", " ", SuperscriptBox["#1", "4"]}], "-", RowBox[{"3", " ", SuperscriptBox["#1", "5"]}], "-", SuperscriptBox["#1", "6"], "+", SuperscriptBox["#1", "7"], "+", SuperscriptBox["#1", "8"]}], "&"}], ",", "2"}], "]"}]]], "Output", CellChangeTimes->{{3.4685370634562597`*^9, 3.4685370920809717`*^9}}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["\[Lambda]", "\[Infinity]"], "[", "\[CapitalGamma]_GradedBigraph", "]"}], ":=", RowBox[{"RootReduce", "[", RowBox[{ RowBox[{ SubscriptBox["\[Rho]", "\[Infinity]"], "[", "\[CapitalGamma]", "]"}], "+", SuperscriptBox[ RowBox[{ SubscriptBox["\[Rho]", "\[Infinity]"], "[", "\[CapitalGamma]", "]"}], RowBox[{"-", "1"}]]}], "]"}]}]], "Input", CellChangeTimes->{{3.468536600615445*^9, 3.468536622388425*^9}, { 3.4685370986927347`*^9, 3.468537116217351*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ SubscriptBox["\[Lambda]", "\[Infinity]"], "[", "\[CapitalGamma]", "]"}]], "Input", CellChangeTimes->{{3.468537057965788*^9, 3.468537063028359*^9}, 3.4685371324493637`*^9}], Cell[BoxData[ RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"4", "-", RowBox[{"32", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"226", " ", SuperscriptBox["#1", "4"]}], "-", RowBox[{"621", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"790", " ", SuperscriptBox["#1", "8"]}], "-", RowBox[{"514", " ", SuperscriptBox["#1", "10"]}], "+", RowBox[{"176", " ", SuperscriptBox["#1", "12"]}], "-", RowBox[{"30", " ", SuperscriptBox["#1", "14"]}], "+", RowBox[{"2", " ", SuperscriptBox["#1", "16"]}]}], "&"}], ",", "4"}], "]"}]], "Output", CellChangeTimes->{3.4685371333771152`*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"\[CapitalPhi]", "[", "m_", "]"}], "[", "x_", "]"}], ":=", RowBox[{"Expand", "[", RowBox[{"2", " ", RowBox[{"ChebyshevT", "[", RowBox[{"m", ",", RowBox[{"x", "/", "2"}]}], "]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.4685373246814537`*^9, 3.468537358737389*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"\[CapitalPhi]", "[", "6", "]"}], "[", RowBox[{"t", "+", SuperscriptBox["t", RowBox[{"-", "1"}]]}], "]"}]], "Input", CellChangeTimes->{{3.4685373487835217`*^9, 3.4685373532174807`*^9}}], Cell[BoxData[ RowBox[{ FractionBox["1", SuperscriptBox["t", "6"]], "+", SuperscriptBox["t", "6"]}]], "Output", CellChangeTimes->{{3.468537353615485*^9, 3.468537359776547*^9}}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"SufficientN", "[", RowBox[{"m_Integer", ",", "\[CapitalGamma]_GradedBigraph"}], "]"}], ":=", RowBox[{ RowBox[{"RootsGreaterThan2Bound", "[", "\[CapitalGamma]", "]"}], RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox[ RowBox[{ RowBox[{"\[CapitalPhi]", "[", "m", "]"}], "[", RowBox[{ SubscriptBox["\[Lambda]", "\[Infinity]"], "[", "\[CapitalGamma]", "]"}], "]"}], "2"], "-", "4"}], ")"}], "/", RowBox[{ "GaloisGroupGrowthConstant", "[", "\[CapitalGamma]", "]"}]}]}]}]], "Input",\ CellChangeTimes->{{3.468536350683571*^9, 3.468536364484428*^9}, { 3.468536565374022*^9, 3.468536695866028*^9}, {3.468537368307556*^9, 3.468537369703485*^9}, 3.468541136022504*^9}], Cell[BoxData[{ RowBox[{ RowBox[{"m", "[", RowBox[{"\[Eta]_", ",", "C_"}], "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"m", "=", "1"}], "}"}], ",", RowBox[{ RowBox[{"While", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[CapitalPhi]", "[", "m", "]"}], "[", "\[Eta]", "]"}], "<", "C"}], ",", RowBox[{"++", "m"}]}], "]"}], ";", "m"}]}], "]"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"m", "[", "\[Eta]_", "]"}], ":=", RowBox[{"m", "[", RowBox[{"\[Eta]", ",", "10"}], "]"}]}]}], "Input", CellChangeTimes->{{3.468541169422427*^9, 3.468541254350088*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"m", "[", "2.08", "]"}]], "Input", CellChangeTimes->{{3.468541238535701*^9, 3.4685412403653727`*^9}}], Cell[BoxData["9"], "Output", CellChangeTimes->{{3.468541241125361*^9, 3.468541255865232*^9}}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"\[Eta]", "[", "\[CapitalGamma]_GradedBigraph", "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"n", "=", "0"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{"r", "=", RowBox[{"Length", "[", RowBox[{"Cases", "[", RowBox[{ RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "t", "}"}], ",", RowBox[{"t", "/.", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"A", "[", "\[CapitalGamma]", "]"}], "[", "t", "]"}], "\[Equal]", "0"}], ",", "t"}], "]"}]}]}], "]"}], ",", RowBox[{"c_", "/;", RowBox[{ RowBox[{ RowBox[{"Im", "[", "c", "]"}], "\[Equal]", "0"}], "\[And]", RowBox[{"c", ">", "1"}]}]}]}], "]"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"While", "[", RowBox[{ RowBox[{ RowBox[{"Length", "[", RowBox[{"Cases", "[", RowBox[{ RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "x", "}"}], ",", RowBox[{"x", "/.", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "n", "]"}], "[", "x", "]"}], "\[Equal]", "0"}], ",", "x"}], "]"}]}]}], "]"}], ",", RowBox[{"c_", "/;", RowBox[{ RowBox[{ RowBox[{"Im", "[", "c", "]"}], "\[Equal]", "0"}], "\[And]", RowBox[{"c", ">", "2"}]}]}]}], "]"}], "]"}], "<", "r"}], ",", RowBox[{"++", "n"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"Min", "[", RowBox[{"Cases", "[", RowBox[{ RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "x", "}"}], ",", RowBox[{"x", "/.", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"P", "[", "\[CapitalGamma]", "]"}], "[", "n", "]"}], "[", "x", "]"}], "\[Equal]", "0"}], ",", "x"}], "]"}]}]}], "]"}], ",", RowBox[{"c_", "/;", RowBox[{ RowBox[{ RowBox[{"Im", "[", "c", "]"}], "\[Equal]", "0"}], "\[And]", RowBox[{"c", ">", "2"}]}]}]}], "]"}], "]"}]}]}], "\[IndentingNewLine]", "]"}]}], "\[IndentingNewLine]", "]"}]}]], "Input",\ CellChangeTimes->{{3.468544526781843*^9, 3.46854453234199*^9}, { 3.4685528920801783`*^9, 3.468552915977611*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Eta]", "[", "\[CapitalGamma]", "]"}]], "Input", CellChangeTimes->{{3.4685529190781717`*^9, 3.468552920863031*^9}}], Cell[BoxData[ SqrtBox[ RowBox[{ FractionBox["5", "2"], "+", FractionBox[ SqrtBox["13"], "2"]}]]], "Output", CellChangeTimes->{3.468552921511734*^9}], Cell[BoxData["1"], "Output", CellChangeTimes->{3.468552738959463*^9, 3.468552795404961*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SufficientN", "[", RowBox[{ RowBox[{"m", "[", RowBox[{"\[Eta]", "[", "\[CapitalGamma]", "]"}], "]"}], ",", "\[CapitalGamma]"}], "]"}]], "Input", CellChangeTimes->{{3.468552957467793*^9, 3.468552968699244*^9}}], Cell[BoxData[ RowBox[{ FractionBox["1", RowBox[{"Log", "[", "2", "]"}]], RowBox[{"10", " ", RowBox[{"HomomorphismConstant", "[", RowBox[{"Function", "[", RowBox[{"t$", ",", FractionBox[ RowBox[{ RowBox[{"-", FractionBox["1", SuperscriptBox["t$", "11"]]}], "-", FractionBox["2", SuperscriptBox["t$", "9"]], "-", FractionBox["1", SuperscriptBox["t$", "7"]], "+", FractionBox["2", SuperscriptBox["t$", "5"]], "+", FractionBox["7", SuperscriptBox["t$", "3"]], "+", FractionBox["12", "t$"], "+", RowBox[{"13", " ", "t$"}], "+", RowBox[{"10", " ", SuperscriptBox["t$", "3"]}], "+", RowBox[{"4", " ", SuperscriptBox["t$", "5"]}], "-", RowBox[{"2", " ", SuperscriptBox["t$", "9"]}], "-", RowBox[{"2", " ", SuperscriptBox["t$", "11"]}]}], RowBox[{ RowBox[{"-", FractionBox["2", SuperscriptBox["t$", "11"]]}], "-", FractionBox["2", SuperscriptBox["t$", "9"]], "+", FractionBox["4", SuperscriptBox["t$", "5"]], "+", FractionBox["10", SuperscriptBox["t$", "3"]], "+", FractionBox["13", "t$"], "+", RowBox[{"12", " ", "t$"}], "+", RowBox[{"7", " ", SuperscriptBox["t$", "3"]}], "+", RowBox[{"2", " ", SuperscriptBox["t$", "5"]}], "-", SuperscriptBox["t$", "7"], "-", RowBox[{"2", " ", SuperscriptBox["t$", "9"]}], "-", SuperscriptBox["t$", "11"]}]]}], "]"}], "]"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "4"}], "+", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"9", " ", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"4", "-", RowBox[{"32", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"226", " ", SuperscriptBox["#1", "4"]}], "-", RowBox[{"621", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"790", " ", SuperscriptBox["#1", "8"]}], "-", RowBox[{"514", " ", SuperscriptBox["#1", "10"]}], "+", RowBox[{"176", " ", SuperscriptBox["#1", "12"]}], "-", RowBox[{"30", " ", SuperscriptBox["#1", "14"]}], "+", RowBox[{"2", " ", SuperscriptBox["#1", "16"]}]}], "&"}], ",", "4"}], "]"}]}], "-", RowBox[{"30", " ", SuperscriptBox[ RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"4", "-", RowBox[{"32", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"226", " ", SuperscriptBox["#1", "4"]}], "-", RowBox[{"621", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"790", " ", SuperscriptBox["#1", "8"]}], "-", RowBox[{"514", " ", SuperscriptBox["#1", "10"]}], "+", RowBox[{"176", " ", SuperscriptBox["#1", "12"]}], "-", RowBox[{"30", " ", SuperscriptBox["#1", "14"]}], "+", RowBox[{"2", " ", SuperscriptBox["#1", "16"]}]}], "&"}], ",", "4"}], "]"}], "3"]}], "+", RowBox[{"27", " ", SuperscriptBox[ RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"4", "-", RowBox[{"32", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"226", " ", SuperscriptBox["#1", "4"]}], "-", RowBox[{"621", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"790", " ", SuperscriptBox["#1", "8"]}], "-", RowBox[{"514", " ", SuperscriptBox["#1", "10"]}], "+", RowBox[{"176", " ", SuperscriptBox["#1", "12"]}], "-", RowBox[{"30", " ", SuperscriptBox["#1", "14"]}], "+", RowBox[{"2", " ", SuperscriptBox["#1", "16"]}]}], "&"}], ",", "4"}], "]"}], "5"]}], "-", RowBox[{"9", " ", SuperscriptBox[ RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"4", "-", RowBox[{"32", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"226", " ", SuperscriptBox["#1", "4"]}], "-", RowBox[{"621", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"790", " ", SuperscriptBox["#1", "8"]}], "-", RowBox[{"514", " ", SuperscriptBox["#1", "10"]}], "+", RowBox[{"176", " ", SuperscriptBox["#1", "12"]}], "-", RowBox[{"30", " ", SuperscriptBox["#1", "14"]}], "+", RowBox[{"2", " ", SuperscriptBox["#1", "16"]}]}], "&"}], ",", "4"}], "]"}], "7"]}], "+", SuperscriptBox[ RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"4", "-", RowBox[{"32", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"226", " ", SuperscriptBox["#1", "4"]}], "-", RowBox[{"621", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"790", " ", SuperscriptBox["#1", "8"]}], "-", RowBox[{"514", " ", SuperscriptBox["#1", "10"]}], "+", RowBox[{"176", " ", SuperscriptBox["#1", "12"]}], "-", RowBox[{"30", " ", SuperscriptBox["#1", "14"]}], "+", RowBox[{"2", " ", SuperscriptBox["#1", "16"]}]}], "&"}], ",", "4"}], "]"}], "9"]}], ")"}], "2"]}], ")"}]}]}]], "Output", CellChangeTimes->{{3.468552969997048*^9, 3.468552984978216*^9}, { 3.468553015944129*^9, 3.468553039669056*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", "%", "]"}]], "Input", CellChangeTimes->{{3.468553110415834*^9, 3.468553111065011*^9}}], Cell[BoxData[ RowBox[{"2327.569260894516`", " ", RowBox[{"HomomorphismConstant", "[", RowBox[{"Function", "[", RowBox[{"t$", ",", FractionBox[ RowBox[{ RowBox[{"-", FractionBox["1.`", SuperscriptBox["t$", "11"]]}], "-", FractionBox["2.`", SuperscriptBox["t$", "9"]], "-", FractionBox["1.`", SuperscriptBox["t$", "7"]], "+", FractionBox["2.`", SuperscriptBox["t$", "5"]], "+", FractionBox["7.`", SuperscriptBox["t$", "3"]], "+", FractionBox["12.`", "t$"], "+", RowBox[{"13.`", " ", "t$"}], "+", RowBox[{"10.`", " ", SuperscriptBox["t$", "3"]}], "+", RowBox[{"4.`", " ", SuperscriptBox["t$", "5"]}], "-", RowBox[{"2.`", " ", SuperscriptBox["t$", "9"]}], "-", RowBox[{"2.`", " ", SuperscriptBox["t$", "11"]}]}], RowBox[{ RowBox[{"-", FractionBox["2.`", SuperscriptBox["t$", "11"]]}], "-", FractionBox["2.`", SuperscriptBox["t$", "9"]], "+", FractionBox["4.`", SuperscriptBox["t$", "5"]], "+", FractionBox["10.`", SuperscriptBox["t$", "3"]], "+", FractionBox["13.`", "t$"], "+", RowBox[{"12.`", " ", "t$"}], "+", RowBox[{"7.`", " ", SuperscriptBox["t$", "3"]}], "+", RowBox[{"2.`", " ", SuperscriptBox["t$", "5"]}], "-", RowBox[{"1.`", " ", SuperscriptBox["t$", "7"]}], "-", RowBox[{"2.`", " ", SuperscriptBox["t$", "9"]}], "-", RowBox[{"1.`", " ", SuperscriptBox["t$", "11"]}]}]]}], "]"}], "]"}]}]], "Output", CellChangeTimes->{3.468553111661674*^9}] }, Open ]] }, WindowSize->{1280, 696}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, FrontEndVersion->"7.0 for Mac OS X x86 (32-bit) (February 18, 2009)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[567, 22, 606, 14, 73, "Input"], Cell[CellGroupData[{ Cell[1198, 40, 463, 10, 36, "Print"], Cell[1664, 52, 428, 9, 20, "Print"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[2141, 67, 184, 3, 27, "Input"], Cell[2328, 72, 780, 13, 145, "Output"] }, Open ]], Cell[3123, 88, 227, 4, 27, "Input"], Cell[3353, 94, 816, 20, 27, "Input"], Cell[4172, 116, 977, 25, 27, "Input"], Cell[5152, 143, 1005, 28, 27, "Input"], Cell[6160, 173, 983, 27, 27, "Input"], Cell[7146, 202, 1542, 46, 43, "Input"], Cell[CellGroupData[{ Cell[8713, 252, 279, 8, 27, "Input"], Cell[8995, 262, 449, 12, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[9481, 279, 221, 6, 27, "Input"], Cell[9705, 287, 439, 13, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[10181, 305, 197, 5, 27, "Input"], Cell[10381, 312, 468, 14, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[10886, 331, 703, 21, 27, "Input"], Cell[11592, 354, 294, 5, 27, "Output"] }, Open ]], Cell[11901, 362, 124, 2, 27, "Input"], Cell[12028, 366, 1617, 43, 51, "Input"], Cell[CellGroupData[{ Cell[13670, 413, 189, 4, 27, "Input"], Cell[13862, 419, 618, 19, 53, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[14517, 443, 300, 8, 33, "Input"], Cell[14820, 453, 678, 22, 48, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[15535, 480, 990, 31, 33, "Input"], Cell[16528, 513, 214, 3, 27, "Output"] }, Open ]], Cell[16757, 519, 937, 26, 33, "Input"], Cell[CellGroupData[{ Cell[17719, 549, 235, 6, 27, "Input"], Cell[17957, 557, 381, 10, 33, "Output"] }, Open ]], Cell[18353, 570, 507, 13, 27, "Input"], Cell[18863, 585, 575, 15, 27, "Input"], Cell[CellGroupData[{ Cell[19463, 604, 162, 3, 27, "Input"], Cell[19628, 609, 370, 10, 51, "Output"] }, Open ]], Cell[20013, 622, 643, 16, 43, "Input"], Cell[CellGroupData[{ Cell[20681, 642, 163, 3, 27, "Input"], Cell[20847, 647, 503, 14, 51, "Output"] }, Open ]], Cell[21365, 664, 1149, 28, 27, "Input"], Cell[CellGroupData[{ Cell[22539, 696, 214, 4, 27, "Input"], Cell[22756, 702, 226, 3, 27, "Output"] }, Open ]], Cell[22997, 708, 1180, 29, 27, "Input"], Cell[CellGroupData[{ Cell[24202, 741, 216, 4, 27, "Input"], Cell[24421, 747, 256, 3, 27, "Output"] }, Open ]], Cell[24692, 753, 350, 8, 49, "Input"], Cell[CellGroupData[{ Cell[25067, 765, 152, 2, 27, "Input"], Cell[25222, 769, 192, 3, 27, "Output"] }, Open ]], Cell[25429, 775, 282, 7, 27, "Input"], Cell[CellGroupData[{ Cell[25736, 786, 226, 6, 27, "Input"], Cell[25965, 794, 141, 2, 27, "Output"] }, Open ]], Cell[26121, 799, 722, 21, 27, "Input"], Cell[CellGroupData[{ Cell[26868, 824, 132, 2, 27, "Input"], Cell[27003, 828, 194, 4, 27, "Output"] }, Open ]], Cell[27212, 835, 1035, 29, 27, "Input"], Cell[CellGroupData[{ Cell[28272, 868, 149, 2, 27, "Input"], Cell[28424, 872, 192, 3, 27, "Output"] }, Open ]], Cell[28631, 878, 2491, 58, 66, "Input"], Cell[CellGroupData[{ Cell[31147, 940, 216, 4, 27, "Input"], Cell[31366, 946, 438, 6, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[31841, 957, 168, 3, 27, "Input"], Cell[32012, 962, 74, 1, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[32123, 968, 127, 2, 27, "Input"], Cell[32253, 972, 154, 3, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[32444, 980, 522, 13, 27, "Input"], Cell[32969, 995, 460, 7, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[33466, 1007, 348, 11, 27, "Input"], Cell[33817, 1020, 77, 1, 27, "Output"] }, Open ]], Cell[33909, 1024, 1177, 34, 33, "Input"], Cell[CellGroupData[{ Cell[35111, 1062, 155, 2, 27, "Input"], Cell[35269, 1066, 71, 1, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[35377, 1072, 276, 8, 33, "Input"], Cell[35656, 1082, 1813, 61, 86, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[37506, 1148, 422, 13, 33, "Input"], Cell[37931, 1163, 1328, 44, 48, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[39296, 1212, 336, 6, 27, "Input"], Cell[39635, 1220, 772, 23, 48, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[40444, 1248, 129, 2, 27, "Input"], Cell[40576, 1252, 436, 14, 48, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[41049, 1271, 487, 14, 27, "Input"], Cell[41539, 1287, 652, 11, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[42228, 1303, 470, 14, 33, "Input"], Cell[42701, 1319, 1383, 45, 48, "Output"] }, Open ]], Cell[44099, 1367, 470, 10, 49, "Input"], Cell[44572, 1379, 636, 17, 53, "Input"], Cell[CellGroupData[{ Cell[45233, 1400, 96, 2, 27, "Input"], Cell[45332, 1404, 627, 21, 48, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[45996, 1430, 601, 18, 27, "Input"], Cell[46600, 1450, 123, 2, 27, "Output"] }, Open ]], Cell[46738, 1455, 903, 23, 27, "Input"], Cell[CellGroupData[{ Cell[47666, 1482, 377, 5, 27, "Input"], Cell[48046, 1489, 73, 1, 27, "Output"] }, Open ]], Cell[48134, 1493, 478, 8, 27, "Input"], Cell[CellGroupData[{ Cell[48637, 1505, 377, 5, 27, "Input"], Cell[49017, 1512, 71, 1, 27, "Output"] }, Open ]], Cell[49103, 1516, 167, 3, 27, "Input"], Cell[49273, 1521, 781, 23, 27, "Input"], Cell[CellGroupData[{ Cell[50079, 1548, 174, 4, 27, "Input"], Cell[50256, 1554, 585, 17, 38, "Output"] }, Open ]], Cell[50856, 1574, 551, 15, 33, "Input"], Cell[CellGroupData[{ Cell[51432, 1593, 205, 5, 27, "Input"], Cell[51640, 1600, 695, 21, 33, "Output"] }, Open ]], Cell[52350, 1624, 338, 9, 27, "Input"], Cell[CellGroupData[{ Cell[52713, 1637, 232, 6, 33, "Input"], Cell[52948, 1645, 186, 5, 48, "Output"] }, Open ]], Cell[53149, 1653, 780, 20, 33, "Input"], Cell[53932, 1675, 666, 21, 43, "Input"], Cell[CellGroupData[{ Cell[54623, 1700, 125, 2, 27, "Input"], Cell[54751, 1704, 94, 1, 27, "Output"] }, Open ]], Cell[54860, 1708, 2917, 78, 103, "Input"], Cell[CellGroupData[{ Cell[57802, 1790, 141, 2, 27, "Input"], Cell[57946, 1794, 163, 6, 66, "Output"], Cell[58112, 1802, 92, 1, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[58241, 1808, 249, 6, 27, "Input"], Cell[58493, 1816, 6258, 170, 293, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[64788, 1991, 120, 2, 27, "Input"], Cell[64911, 1995, 1736, 50, 100, "Output"] }, Open ]] } ] *) (* End of internal cache information *)