(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 1746318, 25760]*) (*NotebookOutlinePosition[ 1746952, 25782]*) (* CellTagsIndexPosition[ 1746908, 25778]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Finiteness for Relative Equilibria of the Four-Body Problem", "Title"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell["\<\ This notebook contains computational algorithms and results which support the \ paper \"Finiteness of relative equilibria of the four-body problem\" by \ Marshall Hampton and Richard Moeckel. The notebook is divided into sections \ explaining the different kinds of computations used to verify the results in \ the paper. The reader can go through the notebook section by section, \ executing the cells in order to check the computations. Some computations were done with the other programs such as Porta, Macaulay2, \ lrs and Mixvol. As explained below, these results have simply been copied \ into this notebook so that further computations based on them can be \ illustrated.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Equations for Relative Equilibria", "Section"], Cell["\<\ First we define some functions to automatically construct the \ Albouy-Chenciner equations for central configurations. For the four-body \ problem, there are six equations for the six mutual distances.\ \>", "Text"], Cell[BoxData[{ \( (*\ Define\ mutual\ distances\ and\ normalized\ variables\ Sij\ *) \ \[IndentingNewLine]r[i_, i_] := 0\), "\[IndentingNewLine]", \(r[i_, j_] := r[j, i] /; j < i\), "\[IndentingNewLine]", \(S[i_, i_] := \ 0\), "\[IndentingNewLine]", \(S[i_, j_]\ := \(\(1/r[i, j]^3 - 1\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\( (*\ Carry\ out\ the\ sum\ in\ equation\ \((10)\)\ of\ section\ 2.1\ *) \)\ \)\), "\[IndentingNewLine]", \(\(\(P[i_, j_, n_]\ := Plus @@ Table[ m[k]*\((S[i, k]*\((r[j, k]^2 - r[i, k]^2 - r[i, j]^2)\) + S[j, k]*\((r[i, k]^2 - r[j, k]^2 - r[i, j]^2)\))\), {k, 1, n}]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\) (*\ Make\ a\ table\ of\ all\ of\ the\ Albouy - Chenciner\ equations\ and\ clear\ the\ denominators\ *) \ \), "\ \[IndentingNewLine]", \(ACeqs[ n_]\ := \ \(\(Table[P[i, j, n], {j, 2, n}, {i, 1, j - 1}] // Flatten\) // Factor\) // Numerator\)}], "Input"], Cell["\<\ Here are the equations written out with the array variables replaced by \ ordinary ones. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(varsub\ = \ {r[1, 2] \[Rule] r12, r[1, 3] \[Rule] r13, r[2, 3] \[Rule] r23, r[1, 4] \[Rule] r14, r[2, 4] \[Rule] r24, r[3, 4] \[Rule] r34, m[1] \[Rule] m1, m[2] \[Rule] m2, m[3] \[Rule] m3, m[4] \[Rule] m4};\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(aceqs\ = \ ACeqs[4] /. varsub\)}], "Input"], Cell[BoxData[ \({\(-m4\)\ r12\^3\ r13\^3\ r14\^3\ r23\^3 + m4\ r12\ r13\^3\ r14\^5\ r23\^3 - m4\ r12\ r13\^3\ r14\^3\ r23\^3\ r24\^2 - m3\ r12\^3\ r13\^3\ r14\^3\ r24\^3 + m3\ r12\ r13\^5\ r14\^3\ r24\^3 - m3\ r12\ r13\^3\ r14\^3\ r23\^2\ r24\^3 - m4\ r12\^3\ r13\^3\ r23\^3\ r24\^3 - m4\ r12\ r13\^3\ r14\^2\ r23\^3\ r24\^3 - m3\ r12\^3\ r14\^3\ r23\^3\ r24\^3 - m3\ r12\ r13\^2\ r14\^3\ r23\^3\ r24\^3 - 2\ m1\ r13\^3\ r14\^3\ r23\^3\ r24\^3 - 2\ m2\ r13\^3\ r14\^3\ r23\^3\ r24\^3 + 2\ m1\ r12\^3\ r13\^3\ r14\^3\ r23\^3\ r24\^3 + 2\ m2\ r12\^3\ r13\^3\ r14\^3\ r23\^3\ r24\^3 + 2\ m3\ r12\^3\ r13\^3\ r14\^3\ r23\^3\ r24\^3 + 2\ m4\ r12\^3\ r13\^3\ r14\^3\ r23\^3\ r24\^3 + m3\ r12\ r14\^3\ r23\^5\ r24\^3 + m4\ r12\ r13\^3\ r23\^3\ r24\^5, \(-m4\)\ r12\^3\ r13\^3\ r14\^3\ r23\ \^3 + m4\ r12\^3\ r13\ r14\^5\ r23\^3 - m4\ r12\^3\ r13\ r14\^3\ r23\^3\ r34\^2 + m2\ r12\^5\ r13\ r14\^3\ r34\^3 - m2\ r12\^3\ r13\^3\ r14\^3\ r34\^3 - m2\ r12\^3\ r13\ r14\^3\ r23\^2\ r34\^3 - m4\ r12\^3\ r13\^3\ r23\^3\ r34\^3 - m4\ r12\^3\ r13\ r14\^2\ r23\^3\ r34\^3 - 2\ m1\ r12\^3\ r14\^3\ r23\^3\ r34\^3 - 2\ m3\ r12\^3\ r14\^3\ r23\^3\ r34\^3 - m2\ r12\^2\ r13\ r14\^3\ r23\^3\ r34\^3 - m2\ r13\^3\ r14\^3\ r23\^3\ r34\^3 + 2\ m1\ r12\^3\ r13\^3\ r14\^3\ r23\^3\ r34\^3 + 2\ m2\ r12\^3\ r13\^3\ r14\^3\ r23\^3\ r34\^3 + 2\ m3\ r12\^3\ r13\^3\ r14\^3\ r23\^3\ r34\^3 + 2\ m4\ r12\^3\ r13\^3\ r14\^3\ r23\^3\ r34\^3 + m2\ r13\ r14\^3\ r23\^5\ r34\^3 + m4\ r12\^3\ r13\ r23\^3\ r34\^5, \(-m4\)\ r12\^3\ r13\^3\ r23\^3\ r24\ \^3 + m4\ r12\^3\ r13\^3\ r23\ r24\^5 - m4\ r12\^3\ r13\^3\ r23\ r24\^3\ r34\^2 - m4\ r12\^3\ r13\^3\ r23\^3\ r34\^3 - m4\ r12\^3\ r13\^3\ r23\ r24\^2\ r34\^3 - 2\ m2\ r12\^3\ r13\^3\ r24\^3\ r34\^3 - 2\ m3\ r12\^3\ r13\^3\ r24\^3\ r34\^3 + m1\ r12\^5\ r23\ r24\^3\ r34\^3 - m1\ r12\^3\ r13\^2\ r23\ r24\^3\ r34\^3 - m1\ r12\^2\ r13\^3\ r23\ r24\^3\ r34\^3 + m1\ r13\^5\ r23\ r24\^3\ r34\^3 - m1\ r12\^3\ r23\^3\ r24\^3\ r34\^3 - m1\ r13\^3\ r23\^3\ r24\^3\ r34\^3 + 2\ m1\ r12\^3\ r13\^3\ r23\^3\ r24\^3\ r34\^3 + 2\ m2\ r12\^3\ r13\^3\ r23\^3\ r24\^3\ r34\^3 + 2\ m3\ r12\^3\ r13\^3\ r23\^3\ r24\^3\ r34\^3 + 2\ m4\ r12\^3\ r13\^3\ r23\^3\ r24\^3\ r34\^3 + m4\ r12\^3\ r13\^3\ r23\ r34\^5, m3\ r12\^3\ r13\^5\ r14\ r24\^3 - m3\ r12\^3\ r13\^3\ r14\^3\ r24\^3 - m3\ r12\^3\ r13\^3\ r14\ r24\^3\ r34\^2 + m2\ r12\^5\ r13\^3\ r14\ r34\^3 - m2\ r12\^3\ r13\^3\ r14\^3\ r34\^3 - m2\ r12\^3\ r13\^3\ r14\ r24\^2\ r34\^3 - 2\ m1\ r12\^3\ r13\^3\ r24\^3\ r34\^3 - 2\ m4\ r12\^3\ r13\^3\ r24\^3\ r34\^3 - m3\ r12\^3\ r13\^2\ r14\ r24\^3\ r34\^3 - m2\ r12\^2\ r13\^3\ r14\ r24\^3\ r34\^3 - m3\ r12\^3\ r14\^3\ r24\^3\ r34\^3 - m2\ r13\^3\ r14\^3\ r24\^3\ r34\^3 + 2\ m1\ r12\^3\ r13\^3\ r14\^3\ r24\^3\ r34\^3 + 2\ m2\ r12\^3\ r13\^3\ r14\^3\ r24\^3\ r34\^3 + 2\ m3\ r12\^3\ r13\^3\ r14\^3\ r24\^3\ r34\^3 + 2\ m4\ r12\^3\ r13\^3\ r14\^3\ r24\^3\ r34\^3 + m2\ r13\^3\ r14\ r24\^5\ r34\^3 + m3\ r12\^3\ r14\ r24\^3\ r34\^5, m3\ r12\^3\ r14\^3\ r23\^5\ r24 - m3\ r12\^3\ r14\^3\ r23\^3\ r24\^3 - m3\ r12\^3\ r14\^3\ r23\^3\ r24\ r34\^2 - 2\ m2\ r12\^3\ r14\^3\ r23\^3\ r34\^3 - 2\ m4\ r12\^3\ r14\^3\ r23\^3\ r34\^3 - m3\ r12\^3\ r14\^3\ r23\^2\ r24\ r34\^3 + m1\ r12\^5\ r23\^3\ r24\ r34\^3 - m1\ r12\^3\ r14\^2\ r23\^3\ r24\ r34\^3 - m1\ r12\^2\ r14\^3\ r23\^3\ r24\ r34\^3 + m1\ r14\^5\ r23\^3\ r24\ r34\^3 - m3\ r12\^3\ r14\^3\ r24\^3\ r34\^3 - m1\ r12\^3\ r23\^3\ r24\^3\ r34\^3 - m1\ r14\^3\ r23\^3\ r24\^3\ r34\^3 + 2\ m1\ r12\^3\ r14\^3\ r23\^3\ r24\^3\ r34\^3 + 2\ m2\ r12\^3\ r14\^3\ r23\^3\ r24\^3\ r34\^3 + 2\ m3\ r12\^3\ r14\^3\ r23\^3\ r24\^3\ r34\^3 + 2\ m4\ r12\^3\ r14\^3\ r23\^3\ r24\^3\ r34\^3 + m3\ r12\^3\ r14\^3\ r24\ r34\^5, \(-2\)\ m3\ r13\^3\ r14\^3\ r23\^3\ \ r24\^3 - 2\ m4\ r13\^3\ r14\^3\ r23\^3\ r24\^3 + m2\ r13\^3\ r14\^3\ r23\^5\ r34 - m2\ r13\^3\ r14\^3\ r23\^3\ r24\^2\ r34 - m2\ r13\^3\ r14\^3\ r23\^2\ r24\^3\ r34 + m1\ r13\^5\ r23\^3\ r24\^3\ r34 - m1\ r13\^3\ r14\^2\ r23\^3\ r24\^3\ r34 - m1\ r13\^2\ r14\^3\ r23\^3\ r24\^3\ r34 + m1\ r14\^5\ r23\^3\ r24\^3\ r34 + m2\ r13\^3\ r14\^3\ r24\^5\ r34 - m2\ r13\^3\ r14\^3\ r23\^3\ r34\^3 - m2\ r13\^3\ r14\^3\ r24\^3\ r34\^3 - m1\ r13\^3\ r23\^3\ r24\^3\ r34\^3 - m1\ r14\^3\ r23\^3\ r24\^3\ r34\^3 + 2\ m1\ r13\^3\ r14\^3\ r23\^3\ r24\^3\ r34\^3 + 2\ m2\ r13\^3\ r14\^3\ r23\^3\ r24\^3\ r34\^3 + 2\ m3\ r13\^3\ r14\^3\ r23\^3\ r24\^3\ r34\^3 + 2\ m4\ r13\^3\ r14\^3\ r23\^3\ r24\^3\ r34\^3}\)], "Output"] }, Open ]], Cell["Next we enter in the three Dziobek equations.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Equations\ \((12)\)\ from\ section\ 2.2\ \ *) \)\(\[IndentingNewLine]\)\(dzeqs\ = \ \({S[1, 2] S[3, 4] - S[1, 4] S[2, 3], S[1, 2] S[3, 4] - S[1, 3] S[2, 4], S[1, 3] S[2, 4] - S[1, 4] S[2, 3]} /. varsub // Factor\) // Numerator\)\)\)], "Input"], Cell[BoxData[ \({r14\^3\ r23\^3 - r12\^3\ r14\^3\ r23\^3 - r12\^3\ r34\^3 + r12\^3\ r14\^3\ r34\^3 + r12\^3\ r23\^3\ r34\^3 - r14\^3\ r23\^3\ r34\^3, r13\^3\ r24\^3 - r12\^3\ r13\^3\ r24\^3 - r12\^3\ r34\^3 + r12\^3\ r13\^3\ r34\^3 + r12\^3\ r24\^3\ r34\^3 - r13\^3\ r24\^3\ r34\^3, r14\^3\ r23\^3 - r13\^3\ r14\^3\ r23\^3 - r13\^3\ r24\^3 + r13\^3\ r14\^3\ r24\^3 + r13\^3\ r23\^3\ r24\^3 - r14\^3\ r23\^3\ r24\^3}\)], "Output"] }, Open ]], Cell["\<\ Our main system consists of these nine equations for the six mutual \ distances. \ \>", "Text"], Cell[BoxData[ \(\(acdzeqs\ = \ aceqs~Join~dzeqs;\)\)], "Input"], Cell["\<\ For constructing the upper bound we also use another system of 10 equations \ and 10 unknowns. We begin by entering the system of 11 equations and \ unknowns (13) from section 2.2 The first five are geometrical equations \ arising from the \"Cayley-Menger\" matrix of squared mutual distances. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(CM = \ {{0, 1, 1, 1, 1}, \[IndentingNewLine]{1, 0, r12^2, r13^2, r14^2}, \[IndentingNewLine]{1, r12^2, 0, r23^2, r24^2}, \[IndentingNewLine]{1, r13^2, r23^2, 0, r34^2}, \[IndentingNewLine]{1, r14^2, r24^2, r34^2, 0}};\)\), "\[IndentingNewLine]", \(geomeqs\ = \ CM . {k, m1*z1, m2*z2, m3*z3, m4*z4}\)}], "Input"], Cell[BoxData[ \({m1\ z1 + m2\ z2 + m3\ z3 + m4\ z4, k + m2\ r12\^2\ z2 + m3\ r13\^2\ z3 + m4\ r14\^2\ z4, k + m1\ r12\^2\ z1 + m3\ r23\^2\ z3 + m4\ r24\^2\ z4, k + m1\ r13\^2\ z1 + m2\ r23\^2\ z2 + m4\ r34\^2\ z4, k + m1\ r14\^2\ z1 + m2\ r24\^2\ z2 + m3\ r34\^2\ z3}\)], "Output"] }, Open ]], Cell["The other six are essentially due to Dziobek.", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(moredzeqs\ = \ \(\(Table[ S[i, j] - z[i]*z[j], {j, 2, 4}, {i, 1, j - 1}] // Flatten\) // Factor\) // Numerator;\)\), "\[IndentingNewLine]", \(\(varsub2\ = \ {z[1] \[Rule] z1, z[2] \[Rule] z2, z[3] \[Rule] z3, z[4] \[Rule] z4};\)\), "\[IndentingNewLine]", \(moredzeqs\ = \ \(moredzeqs /. varsub\) /. varsub2\)}], "Input"], Cell[BoxData[ \({1 - r12\^3 - r12\^3\ z1\ z2, 1 - r13\^3 - r13\^3\ z1\ z3, 1 - r23\^3 - r23\^3\ z2\ z3, 1 - r14\^3 - r14\^3\ z1\ z4, 1 - r24\^3 - r24\^3\ z2\ z4, 1 - r34\^3 - r34\^3\ z3\ z4}\)], "Output"] }, Open ]], Cell["\<\ The system of 10 equations and ten unknowns is obtained by eliminating k. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(geomeqsnok\ = \ {geomeqs[\([1]\)], geomeqs[\([2]\)] - geomeqs[\([5]\)], geomeqs[\([3]\)] - geomeqs[\([5]\)], geomeqs[\([4]\)] - geomeqs[\([5]\)]};\)\), "\[IndentingNewLine]", \(teneqs\ = \ geomeqsnok~Join~moredzeqs\)}], "Input"], Cell[BoxData[ \({m1\ z1 + m2\ z2 + m3\ z3 + m4\ z4, \(-m1\)\ r14\^2\ z1 + m2\ r12\^2\ z2 - m2\ r24\^2\ z2 + m3\ r13\^2\ z3 - m3\ r34\^2\ z3 + m4\ r14\^2\ z4, m1\ r12\^2\ z1 - m1\ r14\^2\ z1 - m2\ r24\^2\ z2 + m3\ r23\^2\ z3 - m3\ r34\^2\ z3 + m4\ r24\^2\ z4, m1\ r13\^2\ z1 - m1\ r14\^2\ z1 + m2\ r23\^2\ z2 - m2\ r24\^2\ z2 - m3\ r34\^2\ z3 + m4\ r34\^2\ z4, 1 - r12\^3 - r12\^3\ z1\ z2, 1 - r13\^3 - r13\^3\ z1\ z3, 1 - r23\^3 - r23\^3\ z2\ z3, 1 - r14\^3 - r14\^3\ z1\ z4, 1 - r24\^3 - r24\^3\ z2\ z4, 1 - r34\^3 - r34\^3\ z3\ z4}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Newton Polytopes", "Section"], Cell[TextData[{ "In this section we find show the ", StyleBox["Mathematica", FontSlant->"Italic"], " functions used to compute the Newton polytopes of our equations and their \ Minkowski sums. The ", StyleBox["Mathematica", FontSlant->"Italic"], " part just involves reading off all the exponent vectors and adding them. \ For the real work of finding the convex hulls we used Porta 1.3.2. The \ final result was verified using lrs 4.1." }], "Text"], Cell[CellGroupData[{ Cell["Exponent Lists", "Subsection"], Cell[TextData[{ "First we have some simple functions based on ", StyleBox["Mathematica's ", FontSlant->"Italic"], StyleBox["built-in Exponent command. Essentially a polynomial which has \ been completely expanded is represented in ", FontVariations->{"CompatibilityType"->0}], StyleBox["Mathematica", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[" as a list of its terms (with a \"Head\" of \"Plus\" in front). \ We also make use of a list of the variables whose exponents we want to read \ off. The first function below is intended to act on a single term of a \ polynomials. It applies ", FontVariations->{"CompatibilityType"->0}], StyleBox["Mathematica", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["'s Exponent command to the monomial for each variable in the \ variable list thereby producing a list of the exponents for all the \ variables. For a polynomial, we just to this for every term to get a list of \ lists of exponents. The Union command eliminates redundancies arising from \ terms which involve the same powers of the variables with different \ coefficients.", FontVariations->{"CompatibilityType"->0}] }], "Text"], Cell[BoxData[{ \(GetExponents[mon_, vars_]\ := \ Map[Exponent[mon, #] &, vars]\), "\[IndentingNewLine]", \(GetAllExponents[poly_, vars_] := \[IndentingNewLine]Table[ GetExponents[poly[\([i]\)], vars], {i, 1, Length[poly]}] // Union\)}], "Input"], Cell["\<\ Here is the exponent list for the Albouy-Chenciner-Dziobek system. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(vars4bp\ = \ {r12, r13, r14, r23, r24, r34};\)\), "\[IndentingNewLine]", \(explist9\ = \ Map[GetAllExponents[#, vars4bp] &, acdzeqs]\)}], "Input"], Cell[BoxData[ \({{{0, 3, 3, 3, 3, 0}, {1, 0, 3, 5, 3, 0}, {1, 2, 3, 3, 3, 0}, {1, 3, 0, 3, 5, 0}, {1, 3, 2, 3, 3, 0}, {1, 3, 3, 2, 3, 0}, {1, 3, 3, 3, 2, 0}, {1, 3, 5, 3, 0, 0}, {1, 5, 3, 0, 3, 0}, {3, 0, 3, 3, 3, 0}, {3, 3, 0, 3, 3, 0}, {3, 3, 3, 0, 3, 0}, {3, 3, 3, 3, 0, 0}, {3, 3, 3, 3, 3, 0}}, {{0, 1, 3, 5, 0, 3}, {0, 3, 3, 3, 0, 3}, {2, 1, 3, 3, 0, 3}, {3, 0, 3, 3, 0, 3}, {3, 1, 0, 3, 0, 5}, {3, 1, 2, 3, 0, 3}, {3, 1, 3, 2, 0, 3}, {3, 1, 3, 3, 0, 2}, {3, 1, 5, 3, 0, 0}, {3, 3, 0, 3, 0, 3}, {3, 3, 3, 0, 0, 3}, {3, 3, 3, 3, 0, 0}, {3, 3, 3, 3, 0, 3}, {5, 1, 3, 0, 0, 3}}, {{0, 3, 0, 3, 3, 3}, {0, 5, 0, 1, 3, 3}, {2, 3, 0, 1, 3, 3}, {3, 0, 0, 3, 3, 3}, {3, 2, 0, 1, 3, 3}, {3, 3, 0, 0, 3, 3}, {3, 3, 0, 1, 0, 5}, {3, 3, 0, 1, 2, 3}, {3, 3, 0, 1, 3, 2}, {3, 3, 0, 1, 5, 0}, {3, 3, 0, 3, 0, 3}, {3, 3, 0, 3, 3, 0}, {3, 3, 0, 3, 3, 3}, {5, 0, 0, 1, 3, 3}}, {{0, 3, 1, 0, 5, 3}, {0, 3, 3, 0, 3, 3}, {2, 3, 1, 0, 3, 3}, {3, 0, 1, 0, 3, 5}, {3, 0, 3, 0, 3, 3}, {3, 2, 1, 0, 3, 3}, {3, 3, 0, 0, 3, 3}, {3, 3, 1, 0, 2, 3}, {3, 3, 1, 0, 3, 2}, {3, 3, 3, 0, 0, 3}, {3, 3, 3, 0, 3, 0}, {3, 3, 3, 0, 3, 3}, {3, 5, 1, 0, 3, 0}, {5, 3, 1, 0, 0, 3}}, {{0, 0, 3, 3, 3, 3}, {0, 0, 5, 3, 1, 3}, {2, 0, 3, 3, 1, 3}, {3, 0, 0, 3, 3, 3}, {3, 0, 2, 3, 1, 3}, {3, 0, 3, 0, 1, 5}, {3, 0, 3, 0, 3, 3}, {3, 0, 3, 2, 1, 3}, {3, 0, 3, 3, 0, 3}, {3, 0, 3, 3, 1, 2}, {3, 0, 3, 3, 3, 0}, {3, 0, 3, 3, 3, 3}, {3, 0, 3, 5, 1, 0}, {5, 0, 0, 3, 1, 3}}, {{0, 0, 3, 3, 3, 3}, {0, 0, 5, 3, 3, 1}, {0, 2, 3, 3, 3, 1}, {0, 3, 0, 3, 3, 3}, {0, 3, 2, 3, 3, 1}, {0, 3, 3, 0, 3, 3}, {0, 3, 3, 0, 5, 1}, {0, 3, 3, 2, 3, 1}, {0, 3, 3, 3, 0, 3}, {0, 3, 3, 3, 2, 1}, {0, 3, 3, 3, 3, 0}, {0, 3, 3, 3, 3, 3}, {0, 3, 3, 5, 0, 1}, {0, 5, 0, 3, 3, 1}}, {{0, 0, 3, 3, 0, 0}, {0, 0, 3, 3, 0, 3}, {3, 0, 0, 0, 0, 3}, {3, 0, 0, 3, 0, 3}, {3, 0, 3, 0, 0, 3}, {3, 0, 3, 3, 0, 0}}, {{0, 3, 0, 0, 3, 0}, {0, 3, 0, 0, 3, 3}, {3, 0, 0, 0, 0, 3}, {3, 0, 0, 0, 3, 3}, {3, 3, 0, 0, 0, 3}, {3, 3, 0, 0, 3, 0}}, {{0, 0, 3, 3, 0, 0}, {0, 0, 3, 3, 3, 0}, {0, 3, 0, 0, 3, 0}, {0, 3, 0, 3, 3, 0}, {0, 3, 3, 0, 3, 0}, {0, 3, 3, 3, 0, 0}}}\)], "Output"] }, Open ]], Cell["\<\ The Albouy-Chenciner equations have 14 essentially different monomials and \ the Dziobek equations have 6. These will be the initial numbers of \ vertices in the initial Newton polytopes, but some of them may not be extreme \ points on the convex hull. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[Length, explist9]\)], "Input"], Cell[BoxData[ \({14, 14, 14, 14, 14, 14, 6, 6, 6}\)], "Output"] }, Open ]], Cell["\<\ In a similar way, we compute the exponent vectors for the auxiliary system of \ 10 equations and 10 unknowns. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tenvars\ = \ {r12, r13, r14, r23, r24, r34, z1, z2, z3, z4};\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(explist10\ = \ Map[GetAllExponents[#, tenvars] &, teneqs]\[IndentingNewLine]\), "\[IndentingNewLine]", \(Map[Length, explist10]\)}], "Input"], Cell[BoxData[ \({{{0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0}}, {{0, 0, 0, 0, 0, 2, 0, 0, 1, 0}, {0, 0, 0, 0, 2, 0, 0, 1, 0, 0}, {0, 0, 2, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 2, 0, 0, 0, 1, 0, 0, 0}, {0, 2, 0, 0, 0, 0, 0, 0, 1, 0}, {2, 0, 0, 0, 0, 0, 0, 1, 0, 0}}, {{0, 0, 0, 0, 0, 2, 0, 0, 1, 0}, {0, 0, 0, 0, 2, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 2, 0, 0, 1, 0, 0}, {0, 0, 0, 2, 0, 0, 0, 0, 1, 0}, {0, 0, 2, 0, 0, 0, 1, 0, 0, 0}, {2, 0, 0, 0, 0, 0, 1, 0, 0, 0}}, {{0, 0, 0, 0, 0, 2, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 2, 0, 0, 1, 0}, {0, 0, 0, 0, 2, 0, 0, 1, 0, 0}, {0, 0, 0, 2, 0, 0, 0, 1, 0, 0}, {0, 0, 2, 0, 0, 0, 1, 0, 0, 0}, {0, 2, 0, 0, 0, 0, 1, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {3, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {3, 0, 0, 0, 0, 0, 1, 1, 0, 0}}, {{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 3, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 3, 0, 0, 0, 0, 1, 0, 1, 0}}, {{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 3, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 3, 0, 0, 0, 1, 1, 0}}, {{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 3, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 3, 0, 0, 0, 1, 0, 0, 1}}, {{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 3, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 3, 0, 0, 1, 0, 1}}, {{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 3, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 3, 0, 0, 1, 1}}}\)], "Output"], Cell[BoxData[ \({4, 6, 6, 6, 3, 3, 3, 3, 3, 3}\)], "Output"] }, Open ]], Cell["\<\ At this point the exponent list explist10 was put into a data file and edited \ to conform with the syntax requirements of the program Mixvol. The resulting \ file, called teneqs.sites was given as input to that program which \ computed the mixed volume of the 10 polytopes. The result was 25380 as \ reported in section 6 of the paper. After this we have no further need for \ the system of 10 equations or its exponent lists. \ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Partial Minkowski Sums", "Subsection"], Cell["\<\ A key step in the method is to find the Minkowski sum of the 9 Newton \ polytopes of the Albouy-Chenciner-Dziobek system. The naive way to find a \ Minkowski sum is just to take all possible sums of nine vertices, one from \ each of the Newton polytopes. In practice, we had to go through cycles of \ taking Minkowski sums of a few of the 9 Newton polytopes, finding a more \ efficient representation of the convex hull using Porta and then adding in \ more of the Newton polytopes. In this way we could succeed in finding \ separate Minkowski sums for the six Albouy-Chenciner equations and for the \ three Dziobek equations. To add these two together required some tricks \ which will be handled in the next subsection. \ \>", "Text"], Cell["\<\ We begin with basic functios to add a single vertex to a list of vertices and \ to add two lists of vertices together. The second function is a primitive \ Minkowski sum for two polytopes but the vertex list will be unnecessarily \ long. One needs a program like Porta to find the true extreme points of the \ convex hull. \ \>", "Text"], Cell[BoxData[{ \(AddToAll[vect_, vlist_]\ := Map[\((\ vect + #)\) &, vlist]\), "\[IndentingNewLine]", \(MinkSum[vlist_, wlist_]\ := \ \(Map[AddToAll[#, wlist] &, vlist] // Flatten[#, 1] &\) // Union\)}], "Input"], Cell["\<\ Here is the primitive Minkowski sum for the three Dziobek equations. We get a \ list of 132 vertices. \ \>", "Text"], Cell[BoxData[{ \(dzmink\ = \ MinkSum[MinkSum[explist9[\([7]\)], explist9[\([8]\)]], explist9[\([9]\)]]\), "\[IndentingNewLine]", \(Length[dzmink]\)}], "Input"], Cell[TextData[{ "We put this list in a file named \"dzmink.poi\" and edited it to conform \ with the syntax of the program Porta 1.3.2. Running the command ", StyleBox["traf -l dzmink.poi ", FontWeight->"Bold"], "produces a file called \"dzmink.poi.ieq\" containing a minimal list of \ inequalities in six variables x1, ... , x6 defining the convex hull . \ Running the command ", StyleBox["traf -l dzmink.poi.ieq ", FontWeight->"Bold"], StyleBox["produces yet another file \"dzmink.poi.ieq.poi which contains the \ true vertices of the Minkowski sum polytope. To avoid having to deal with \ these files here, we will just copy the results, showing the commands in \ comments", FontVariations->{"CompatibilityType"->0}] }], "Text"], Cell[BoxData[{ \( (*\ Write\ the\ raw\ vertices\ to\ a\ file\ *) \[IndentingNewLine] (*\ Put[dzmink, "\"]\ *) \ \[IndentingNewLine]\[IndentingNewLine] (*\ Read\ in\ the\ results\ from\ porta, \ after\ editing\ the\ output\ files\ to\ Mathematica\ syntax\ *) \ \[IndentingNewLine] (*\ dzconv\ = \ ReadList["\", {Number, Number, Number, Number, Number, Number}]*) (*\ dzineq\ = \ ReadList["\", Expression]*) \[IndentingNewLine]\[IndentingNewLine]\(dzconv\ = \ \ {{3, 6, 0, 0, 6, 3}, {0, 3, 6, 6, 6, 0}, {3, 6, 0, 0, 6, 6}, {3, 6, 0, 6, 6, 6}, {6, 3, 0, 0, 6, 6}, {6, 3, 0, 6, 6, 6}, {0, 3, 6, 6, 3, 0}, {0, 6, 3, 3, 6, 0}, {0, 3, 6, 6, 6, 6}, {0, 6, 3, 6, 6, 0}, {0, 6, 3, 6, 6, 6}, {0, 6, 6, 3, 6, 0}, {0, 6, 6, 3, 6, 6}, {0, 6, 6, 6, 3, 0}, {6, 3, 6, 0, 6, 6}, {0, 3, 6, 6, 3, 6}, {0, 6, 3, 3, 6, 6}, {0, 6, 6, 6, 3, 6}, {6, 6, 0, 0, 3, 6}, {6, 6, 0, 6, 3, 6}, {6, 6, 6, 0, 3, 6}, {6, 0, 3, 3, 0, 6}, {6, 0, 3, 3, 6, 6}, {6, 0, 3, 6, 0, 6}, {6, 0, 3, 6, 6, 6}, {3, 0, 6, 6, 0, 3}, {3, 0, 6, 6, 6, 3}, {3, 0, 6, 6, 0, 6}, {3, 0, 6, 6, 6, 6}, {6, 0, 6, 3, 0, 6}, {6, 0, 6, 3, 6, 6}, {3, 6, 6, 6, 0, 3}, {3, 6, 6, 6, 0, 6}, {6, 6, 3, 3, 0, 6}, {6, 6, 3, 6, 0, 6}, {6, 6, 6, 3, 0, 6}, {6, 0, 6, 6, 0, 3}, {6, 0, 6, 6, 6, 3}, {6, 6, 0, 0, 6, 3}, {6, 6, 0, 6, 6, 3}, {6, 6, 6, 6, 0, 3}, {6, 3, 6, 6, 6, 0}, {6, 3, 6, 6, 3, 0}, {6, 6, 3, 3, 6, 0}, {3, 6, 6, 0, 6, 6}, {6, 6, 3, 6, 6, 0}, {6, 6, 6, 0, 6, 3}, {6, 6, 6, 3, 6, 0}, {3, 6, 0, 6, 6, 3}, {3, 6, 6, 0, 6, 3}, {6, 3, 0, 0, 3, 6}, {6, 3, 0, 6, 3, 6}, {6, 3, 6, 0, 3, 6}, {6, 6, 6, 6, 3, 0}};\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(dzineq\ = \ {x1 + x2 + x5 + x6 \[LessEqual] 21, \(-x1\) - x2 - x3 \[LessEqual] \(-9\), \(-x1\) - x2 - x4 \[LessEqual] \(-9\), x3 \[LessEqual] 6, \(-x1\) - x4 - x5 \[LessEqual] \(-9\), x4 \[LessEqual] 6, \(-x1\) - x3 - x5 \[LessEqual] \(-9\), x6 \[LessEqual] 6, x5 \[LessEqual] 6, \(-x6\) \[LessEqual] 0, \(-x2\) - x3 - x6 \[LessEqual] \(-9\), \(-x2\) - x4 - x6 \[LessEqual] \(-9\), \(-x3\) - x5 - x6 \[LessEqual] \(-9\), \(-x4\) - x5 - x6 \[LessEqual] \(-9\), x2 + x3 + x4 + x5 \[LessEqual] 21, \(-x1\) \[LessEqual] 0, x2 \[LessEqual] 6, \(-x2\) \[LessEqual] 0, \(-x5\) \[LessEqual] 0, x1 + x3 + x4 + x6 \[LessEqual] 21, x1 \[LessEqual] 6, \(-x3\) \[LessEqual] 0, \(-x4\) \[LessEqual] 0, x1 + x2 + x3 + x4 + x5 + x6 \[LessEqual] 27};\)\)}], "Input"], Cell["\<\ This is the description of the polytope P2 in section 4.1 of the paper. It \ has 54 vertices and 24 facets. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Length[dzconv]\), "\[IndentingNewLine]", \(Length[dzineq]\)}], "Input"], Cell[BoxData[ \(54\)], "Output"], Cell[BoxData[ \(24\)], "Output"] }, Open ]], Cell["\<\ A similar approach was used for the six Albouy-Chenciner equations. This \ time it is not possible to add all six at once. Instead we had to add three \ of the polytopes, reduce the number of hull vertices with Porta, add in the \ fourth, reduces the hull, etc. Here are the final results. To save space, \ this cell has been kept closed. Select it and open it to see the results. \ In any case, the cell should be selected and executed at this time before \ continuing. \ \>", "Text"], Cell[BoxData[{ \(\(acconv = {{0, 13, 13, 17, 17, 15}, {0, 13, 13, 19, 14, 13}, {0, 10, 15, 17, 17, 13}, {0, 10, 13, 17, 17, 15}, {0, 13, 17, 17, 13, 15}, {0, 13, 17, 19, 10, 13}, {0, 10, 19, 17, 13, 13}, {0, 10, 17, 17, 13, 15}, {0, 13, 17, 17, 10, 15}, {0, 13, 10, 17, 17, 15}, {0, 13, 13, 14, 19, 13}, {0, 13, 13, 17, 17, 12}, {0, 15, 10, 17, 17, 13}, {0, 17, 17, 13, 13, 15}, {0, 17, 17, 15, 10, 13}, {0, 14, 19, 13, 13, 13}, {0, 17, 17, 13, 10, 15}, {0, 17, 13, 13, 17, 15}, {0, 17, 10, 13, 17, 15}, {0, 17, 13, 10, 19, 13}, {0, 19, 10, 13, 17, 13}, {0, 17, 17, 10, 13, 15}, {0, 17, 13, 10, 17, 15}, {0, 17, 17, 10, 15, 13}, {0, 19, 14, 13, 13, 13}, {0, 17, 17, 13, 13, 12}, {0, 17, 13, 13, 17, 12}, {0, 13, 17, 17, 13, 12}, {1, 10, 13, 19, 17, 15}, {1, 10, 13, 21, 14, 13}, {1, 7, 15, 19, 17, 13}, {1, 7, 13, 19, 17, 15}, {1, 10, 17, 19, 13, 15}, {1, 10, 17, 21, 10, 13}, {1, 7, 19, 19, 13, 13}, {1, 7, 17, 19, 13, 15}, {1, 10, 17, 19, 10, 15}, {1, 10, 10, 19, 17, 15}, {1, 13, 10, 17, 19, 15}, {1, 10, 12, 17, 19, 13}, {1, 10, 10, 17, 19, 15}, {1, 13, 10, 14, 21, 13}, {1, 10, 13, 16, 19, 13}, {1, 13, 10, 17, 19, 12}, {1, 13, 7, 17, 19, 15}, {1, 10, 13, 19, 17, 12}, {1, 12, 10, 19, 17, 13}, {1, 15, 7, 17, 19, 13}, {1, 13, 19, 17, 10, 15}, {1, 13, 19, 19, 7, 13}, {1, 10, 21, 17, 10, 13}, {1, 10, 19, 17, 10, 15}, {1, 13, 19, 17, 7, 15}, {1, 10, 17, 19, 13, 12}, {1, 17, 10, 10, 21, 13}, {1, 17, 10, 13, 19, 15}, {1, 17, 7, 13, 19, 15}, {1, 17, 10, 10, 19, 15}, {1, 13, 10, 19, 16, 13}, {3, 13, 13, 15, 19, 12}, {3, 10, 15, 15, 19, 10}, {3, 13, 13, 12, 21, 10}, {3, 13, 13, 15, 19, 9}, {6, 13, 15, 13, 19, 6}, {4, 10, 13, 17, 19, 12}, {4, 7, 15, 17, 19, 10}, {4, 13, 10, 15, 21, 12}, {4, 10, 12, 15, 21, 10}, {4, 13, 10, 12, 23, 10}, {4, 13, 10, 15, 21, 9}, {4, 10, 13, 17, 19, 9}, {1, 17, 19, 13, 10, 15}, {1, 14, 21, 13, 10, 13}, {1, 17, 19, 13, 7, 15}, {1, 17, 19, 10, 10, 15}, {1, 17, 19, 15, 7, 13}, {1, 17, 19, 13, 10, 12}, {1, 13, 19, 17, 10, 12}, {1, 17, 19, 10, 12, 13}, {1, 19, 17, 10, 13, 15}, {1, 16, 19, 10, 13, 13}, {1, 19, 17, 7, 13, 15}, {1, 19, 17, 7, 15, 13}, {1, 19, 7, 13, 19, 13}, {1, 17, 10, 13, 19, 12}, {3, 13, 13, 19, 15, 12}, {3, 13, 13, 21, 12, 10}, {3, 13, 13, 19, 15, 9}, {3, 15, 10, 19, 15, 10}, {6, 15, 13, 19, 13, 6}, {4, 10, 13, 21, 15, 12}, {4, 10, 13, 23, 12, 10}, {4, 13, 10, 19, 17, 9}, {4, 13, 10, 19, 17, 12}, {4, 10, 13, 21, 15, 9}, {4, 12, 10, 21, 15, 10}, {4, 15, 7, 19, 17, 10}, {3, 15, 19, 13, 13, 12}, {3, 12, 21, 13, 13, 10}, {3, 15, 19, 10, 15, 10}, {3, 15, 19, 13, 13, 9}, {6, 13, 19, 13, 15, 6}, {4, 15, 21, 13, 10, 12}, {4, 12, 23, 13, 10, 10}, {4, 15, 21, 13, 10, 9}, {4, 15, 21, 10, 12, 10}, {4, 17, 19, 10, 13, 12}, {4, 17, 19, 7, 15, 10}, {1, 19, 13, 7, 17, 15}, {1, 21, 10, 10, 17, 13}, {1, 19, 13, 10, 17, 15}, {1, 19, 10, 10, 17, 15}, {1, 19, 13, 7, 19, 13}, {1, 19, 17, 12, 10, 13}, {1, 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15, 7, 10}, {21, 10, 7, 15, 4, 15}, {21, 10, 7, 13, 4, 17}, {23, 7, 10, 12, 7, 13}, {21, 7, 4, 10, 13, 17}, {21, 7, 10, 7, 15, 15}, {21, 7, 7, 7, 15, 15}, {21, 12, 7, 7, 15, 10}, {21, 12, 10, 7, 15, 10}, {21, 10, 12, 7, 15, 10}, {21, 7, 12, 7, 15, 10}, {21, 7, 15, 7, 10, 15}, {21, 4, 15, 7, 10, 15}, {21, 7, 15, 7, 7, 15}, {21, 10, 13, 7, 4, 17}, {21, 4, 13, 7, 10, 17}, {21, 7, 10, 4, 15, 15}, {21, 7, 10, 4, 13, 17}, {23, 7, 10, 7, 12, 13}, {23, 10, 7, 7, 12, 13}, {23, 10, 7, 12, 7, 13}, {21, 4, 15, 7, 12, 13}, {21, 7, 15, 12, 7, 10}, {21, 7, 15, 12, 10, 10}, {21, 7, 15, 10, 12, 10}, {21, 7, 15, 7, 12, 10}, {21, 7, 13, 10, 12, 9}, {21, 7, 13, 12, 10, 9}, {21, 7, 12, 4, 15, 13}, {21, 10, 12, 7, 13, 9}, {21, 12, 10, 7, 13, 9}, {21, 10, 12, 13, 7, 9}, {21, 12, 10, 13, 7, 9}, {21, 12, 7, 15, 4, 13}, {23, 7, 12, 10, 7, 13}, {23, 7, 12, 7, 10, 13}, {21, 13, 10, 4, 7, 17}, {21, 15, 7, 10, 7, 15}, {21, 15, 4, 10, 7, 15}, {21, 15, 7, 7, 7, 15}, {21, 13, 4, 10, 7, 17}, {21, 13, 7, 12, 10, 9}, {21, 15, 7, 7, 12, 10}, {21, 15, 7, 10, 12, 10}, {21, 13, 7, 10, 12, 9}, {21, 15, 7, 12, 10, 10}, {21, 15, 7, 12, 7, 10}, {23, 12, 7, 7, 10, 13}, {23, 12, 7, 10, 7, 13}, {21, 15, 4, 12, 7, 13}, {23, 12, 7, 10, 10, 13}};\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(acineq\ = \ {\(-x1\) \[LessEqual] 0, x3\/45 + x5\/45 + x6\/45 \[LessEqual] 1, \(-\(x1\/72\)\) - x2\/72 - x3\/72 - x4\/72 - x5\/72 - x6\/72 \[LessEqual] \(-1\), \(-\(\(2\ x1\)\/35\)\) + \(2\ \ x3\)\/175 + x4\/35 + \(2\ x5\)\/175 + \(2\ x6\)\/175 \[LessEqual] 1, \(-\(x1\/16\)\) + x4\/32 + x6\/32 \[LessEqual] 1, \(-\(\(6\ x1\)\/85\)\) + \(3\ x4\)\/85 + \(2\ x5\)\/85 \ \[LessEqual] 1, \(-\(\(6\ x1\)\/385\)\) + \(4\ x2\)\/385 + \(6\ x3\)\/385 + \ \(9\ x4\)\/385 + \(6\ x5\)\/385 \[LessEqual] 1, \(-\(\(2\ x1\)\/49\)\) + x3\/49 + x4\/49 + x6\/49 \[LessEqual] 1, \(-\(\(6\ x1\)\/85\)\) + \(2\ x4\)\/85 + \(3\ x5\)\/85 \ \[LessEqual] 1, \(-\(x1\/16\)\) + x5\/32 + x6\/32 \[LessEqual] 1, \(-\(\(4\ x1\)\/175\)\) + \(2\ x3\)\/175 + \(2\ x4\)\/175 + x5\/35 + \(2\ x6\)\/175 \[LessEqual] 1, x3\/61 + \(3\ x5\)\/122 + \(3\ x6\)\/122 \[LessEqual] 1, \(-\(x1\/17\)\) + x4\/34 + x5\/34 \[LessEqual] 1, \(-\(x1\/16\)\) + x3\/32 + x6\/32 \[LessEqual] 1, \(3\ x3\)\/122 + x5\/61 + \(3\ x6\)\/122 \[LessEqual] 1, \(-\(\(6\ x1\)\/385\)\) + \(6\ x2\)\/385 + \(9\ x3\)\/385 + \ \(6\ x4\)\/385 + \(4\ x5\)\/385 \[LessEqual] 1, \(-\(\(2\ x1\)\/35\)\) + \(2\ x2\)\/175 + x3\/35 + \(2\ x4\)\/175 + \(2\ x6\)\/175 \[LessEqual] 1, x2\/77 + \(3\ x3\)\/154 + \(3\ x4\)\/154 + x5\/77 \[LessEqual] 1, \(-\(\(2\ x1\)\/35\)\) + \(2\ x2\)\/175 + \(2\ x4\)\/175 + x5\/35 + \(2\ x6\)\/175 \[LessEqual] 1, x2\/45 + x4\/45 + x6\/45 \[LessEqual] 1, x2\/61 + \(3\ x4\)\/122 + \(3\ x6\)\/122 \[LessEqual] 1, \(-\(\(4\ x1\)\/175\)\) + \(2\ x2\)\/175 + x4\/35 + \(2\ x5\)\/175 + \(2\ x6\)\/175 \[LessEqual] 1, \(-\(\(6\ x1\)\/385\)\) + \(6\ x2\)\/385 + \(4\ x3\)\/385 + \ \(6\ x4\)\/385 + \(9\ x5\)\/385 \[LessEqual] 1, \(-\(\(6\ x1\)\/85\)\) + \(2\ x2\)\/85 + \(3\ x3\)\/85 \ \[LessEqual] 1, \(-\(\(4\ x1\)\/175\)\) + \(2\ x2\)\/175 + x3\/35 + \(2\ x5\)\/175 + \(2\ x6\)\/175 \[LessEqual] 1, \(-\(\(2\ x1\)\/49\)\) + x2\/49 + x5\/49 + x6\/49 \[LessEqual] 1, \(-\(x1\/16\)\) + x2\/32 + x6\/32 \[LessEqual] 1, \(3\ x2\)\/122 + x4\/61 + \(3\ x6\)\/122 \[LessEqual] 1, \(-\(\(2\ x1\)\/35\)\) + x2\/35 + \(2\ x3\)\/175 + \(2\ x5\)\/175 + \(2\ x6\)\/175 \ \[LessEqual] 1, \(-\(\(6\ x1\)\/385\)\) + \(9\ x2\)\/385 + \(6\ x3\)\/385 + \ \(4\ x4\)\/385 + \(6\ x5\)\/385 \[LessEqual] 1, \(3\ x2\)\/154 + x3\/77 + x4\/77 + \(3\ x5\)\/154 \[LessEqual] 1, \(-\(\(6\ x1\)\/85\)\) + \(3\ x2\)\/85 + \(2\ x3\)\/85 \ \[LessEqual] 1, \(-\(x1\/17\)\) + x2\/34 + x3\/34 \[LessEqual] 1, \(-\(\(4\ x1\)\/175\)\) + x2\/35 + \(2\ x3\)\/175 + \(2\ x4\)\/175 + \(2\ x6\)\/175 \ \[LessEqual] 1, \(-\(\(2\ x1\)\/45\)\) + x6\/15 \[LessEqual] 1, \(-\(x1\/31\)\) + x4\/62 + \(3\ x6\)\/62 \[LessEqual] 1, \(-\(x1\/47\)\) + x4\/94 + x5\/94 + \(2\ x6\)\/47 \[LessEqual] 1, \(-\(x1\/47\)\) + x3\/94 + x4\/94 + \(2\ x6\)\/47 \[LessEqual] 1, \(-\(x1\/31\)\) + x5\/62 + \(3\ x6\)\/62 \[LessEqual] 1, \(-\(x1\/225\)\) + x2\/75 + x3\/50 + x4\/75 + x5\/50 \[LessEqual] 1, \(2\ x2\)\/167 + \(3\ x3\)\/167 + \(3\ x4\)\/167 + \(3\ \ x5\)\/167 \[LessEqual] 1, \(-\(\(2\ x1\)\/419\)\) + \(4\ x2\)\/419 + \(6\ x3\)\/419 + \ \(7\ x4\)\/419 + \(10\ x5\)\/419 \[LessEqual] 1, \(-\(\(18\ x1\)\/3335\)\) + \(32\ x2\)\/3335 + \(48\ \ x3\)\/3335 + \(54\ x4\)\/3335 + \(81\ x5\)\/3335 \[LessEqual] 1, \(-\(\(4\ x1\)\/945\)\) + \(8\ x2\)\/945 + \(2\ x3\)\/135 + \ \(2\ x4\)\/135 + \(23\ x5\)\/945 + \(2\ x6\)\/945 \[LessEqual] 1, \(-\(x1\/31\)\) + x3\/62 + \(3\ x6\)\/62 \[LessEqual] 1, \(-\(\(6\ x1\)\/1207\)\) + \(18\ x2\)\/1207 + \(27\ x3\)\/1207 \ + \(18\ x4\)\/1207 + \(16\ x5\)\/1207 \[LessEqual] 1, \(-\(\(18\ x1\)\/3335\)\) + \(54\ x2\)\/3335 + \(81\ \ x3\)\/3335 + \(32\ x4\)\/3335 + \(48\ x5\)\/3335 \[LessEqual] 1, \(-\(\(4\ x1\)\/945\)\) + \(2\ x2\)\/135 + \(23\ x3\)\/945 + \ \(8\ x4\)\/945 + \(2\ x5\)\/135 + \(2\ x6\)\/945 \[LessEqual] 1, \(-\(\(6\ x1\)\/1207\)\) + \(18\ x2\)\/1207 + \(16\ x3\)\/1207 \ + \(18\ x4\)\/1207 + \(27\ x5\)\/1207 \[LessEqual] 1, \(3\ x2\)\/167 + \(2\ x3\)\/167 + \(3\ x4\)\/167 + \(3\ \ x5\)\/167 \[LessEqual] 1, \(-\(x1\/225\)\) + x2\/50 + x3\/75 + x4\/50 + x5\/75 \[LessEqual] 1, \(-\(\(18\ x1\)\/3335\)\) + \(48\ x2\)\/3335 + \(32\ \ x3\)\/3335 + \(81\ x4\)\/3335 + \(54\ x5\)\/3335 \[LessEqual] 1, \(-\(\(6\ x1\)\/1207\)\) + \(16\ x2\)\/1207 + \(18\ x3\)\/1207 \ + \(27\ x4\)\/1207 + \(18\ x5\)\/1207 \[LessEqual] 1, \(-\(\(2\ x1\)\/419\)\) + \(6\ x2\)\/419 + \(4\ x3\)\/419 + \ \(10\ x4\)\/419 + \(7\ x5\)\/419 \[LessEqual] 1, x2\/60 + x3\/60 + x4\/60 + x5\/60 \[LessEqual] 1, x2\/77 + x3\/77 + \(3\ x4\)\/154 + \(3\ x5\)\/154 \[LessEqual] 1, \(-\(\(4\ x1\)\/945\)\) + \(2\ x2\)\/135 + \(8\ x3\)\/945 + \ \(23\ x4\)\/945 + \(2\ x5\)\/135 + \(2\ x6\)\/945 \[LessEqual] 1, \(3\ x2\)\/167 + \(3\ x3\)\/167 + \(3\ x4\)\/167 + \(2\ \ x5\)\/167 \[LessEqual] 1, \(-\(x1\/47\)\) + x2\/94 + x5\/94 + \(2\ x6\)\/47 \[LessEqual] 1, \(3\ x2\)\/167 + \(3\ x3\)\/167 + \(2\ x4\)\/167 + \(3\ \ x5\)\/167 \[LessEqual] 1, \(-\(x1\/47\)\) + x2\/94 + x3\/94 + \(2\ x6\)\/47 \[LessEqual] 1, \(-\(\(2\ x1\)\/419\)\) + \(7\ x2\)\/419 + \(10\ x3\)\/419 + \ \(4\ x4\)\/419 + \(6\ x5\)\/419 \[LessEqual] 1, \(-\(x1\/31\)\) + x2\/62 + \(3\ x6\)\/62 \[LessEqual] 1, \(-\(\(6\ x1\)\/1207\)\) + \(27\ x2\)\/1207 + \(18\ x3\)\/1207 \ + \(16\ x4\)\/1207 + \(18\ x5\)\/1207 \[LessEqual] 1, \(-\(\(2\ x1\)\/419\)\) + \(10\ x2\)\/419 + \(7\ x3\)\/419 + \ \(6\ x4\)\/419 + \(4\ x5\)\/419 \[LessEqual] 1, \(-\(\(18\ x1\)\/3335\)\) + \(81\ x2\)\/3335 + \(54\ \ x3\)\/3335 + \(48\ x4\)\/3335 + \(32\ x5\)\/3335 \[LessEqual] 1, \(3\ x2\)\/154 + \(3\ x3\)\/154 + x4\/77 + x5\/77 \[LessEqual] 1, \(-\(\(4\ x1\)\/945\)\) + \(23\ x2\)\/945 + \(2\ x3\)\/135 + \ \(2\ x4\)\/135 + \(8\ x5\)\/945 + \(2\ x6\)\/945 \[LessEqual] 1, \(2\ x3\)\/185 + x4\/37 + \(2\ x5\)\/185 + \(2\ x6\)\/185 \[LessEqual] 1, \(-\(x1\/50\)\) + \(3\ x4\)\/100 + \(3\ x6\)\/100 \[LessEqual] 1, \(2\ x3\)\/179 + \(2\ x4\)\/179 + \(5\ x5\)\/179 + \(2\ \ x6\)\/179 \[LessEqual] 1, x4\/36 + x5\/36 \[LessEqual] 1, x4\/51 + x5\/51 + x6\/51 \[LessEqual] 1, \(5\ x4\)\/159 + \(2\ x5\)\/159 + \(2\ x6\)\/159 \[LessEqual] 1, x3\/51 + x4\/51 + x6\/51 \[LessEqual] 1, \(-\(\(2\ x1\)\/271\)\) + \(9\ x4\)\/271 + \(6\ x5\)\/271 \ \[LessEqual] 1, \(-\(\(2\ x1\)\/389\)\) + \(4\ x2\)\/389 + \(6\ x3\)\/389 + \ \(9\ x4\)\/389 + \(6\ x5\)\/389 \[LessEqual] 1, \(-\(x1\/50\)\) + \(3\ x5\)\/100 + \(3\ x6\)\/100 \[LessEqual] 1, \(2\ x4\)\/159 + \(5\ x5\)\/159 + \(2\ x6\)\/159 \[LessEqual] 1, \(2\ x2\)\/179 + \(5\ x4\)\/179 + \(2\ x5\)\/179 + \(2\ \ x6\)\/179 \[LessEqual] 1, \(-\(\(2\ x1\)\/271\)\) + \(6\ x4\)\/271 + \(9\ x5\)\/271 \ \[LessEqual] 1, \(4\ x2\)\/391 + \(6\ x3\)\/391 + \(6\ x4\)\/391 + \(9\ \ x5\)\/391 \[LessEqual] 1, \(-\(x1\/50\)\) + \(3\ x3\)\/100 + \(3\ x6\)\/100 \[LessEqual] 1, \(2\ x2\)\/185 + x3\/37 + \(2\ x4\)\/185 + \(2\ x6\)\/185 \[LessEqual] 1, \(-\(\(2\ x1\)\/389\)\) + \(6\ x2\)\/389 + \(9\ x3\)\/389 + \ \(6\ x4\)\/389 + \(4\ x5\)\/389 \[LessEqual] 1, \(-\(\(2\ x1\)\/271\)\) + \(6\ x2\)\/271 + \(9\ x3\)\/271 \ \[LessEqual] 1, \(2\ x2\)\/179 + \(5\ x3\)\/179 + \(2\ x5\)\/179 + \(2\ \ x6\)\/179 \[LessEqual] 1, \(2\ x2\)\/159 + \(5\ x3\)\/159 + \(2\ x6\)\/159 \[LessEqual] 1, \(2\ x2\)\/185 + \(2\ x4\)\/185 + x5\/37 + \(2\ x6\)\/185 \[LessEqual] 1, \(-\(\(2\ x1\)\/389\)\) + \(6\ x2\)\/389 + \(4\ x3\)\/389 + \ \(6\ x4\)\/389 + \(9\ x5\)\/389 \[LessEqual] 1, \(6\ x2\)\/391 + \(4\ x3\)\/391 + \(9\ x4\)\/391 + \(6\ \ x5\)\/391 \[LessEqual] 1, \(6\ x2\)\/391 + \(9\ x3\)\/391 + \(4\ x4\)\/391 + \(6\ \ x5\)\/391 \[LessEqual] 1, x2\/51 + x5\/51 + x6\/51 \[LessEqual] 1, x2\/36 + x3\/36 \[LessEqual] 1, x2\/51 + x3\/51 + x6\/51 \[LessEqual] 1, \(5\ x2\)\/179 + \(2\ x3\)\/179 + \(2\ x4\)\/179 + \(2\ \ x6\)\/179 \[LessEqual] 1, \(-\(x1\/50\)\) + \(3\ x2\)\/100 + \(3\ x6\)\/100 \[LessEqual] 1, x2\/37 + \(2\ x3\)\/185 + \(2\ x5\)\/185 + \(2\ x6\)\/185 \ \[LessEqual] 1, \(5\ x2\)\/159 + \(2\ x3\)\/159 + \(2\ x6\)\/159 \[LessEqual] 1, \(-\(\(2\ x1\)\/389\)\) + \(9\ x2\)\/389 + \(6\ x3\)\/389 + \ \(4\ x4\)\/389 + \(6\ x5\)\/389 \[LessEqual] 1, \(-\(\(2\ x1\)\/271\)\) + \(9\ x2\)\/271 + \(6\ x3\)\/271 \ \[LessEqual] 1, \(9\ x2\)\/391 + \(6\ x3\)\/391 + \(6\ x4\)\/391 + \(4\ \ x5\)\/391 \[LessEqual] 1, \(-\(\(2\ x1\)\/79\)\) + x4\/79 + \(4\ x6\)\/79 \[LessEqual] 1, \(-\(\(2\ x1\)\/79\)\) + x5\/79 + \(4\ x6\)\/79 \[LessEqual] 1, \(-\(\(4\ x1\)\/265\)\) + \(2\ x3\)\/265 + \(2\ x4\)\/265 + \ \(2\ x5\)\/265 + \(11\ x6\)\/265 \[LessEqual] 1, \(-\(\(2\ x1\)\/79\)\) + x3\/79 + \(4\ x6\)\/79 \[LessEqual] 1, \(2\ x2\)\/139 + \(3\ x3\)\/139 + \(2\ x4\)\/139 + \(2\ \ x5\)\/139 \[LessEqual] 1, \(-\(\(4\ x1\)\/265\)\) + \(2\ x2\)\/265 + \(2\ x4\)\/265 + \ \(2\ x5\)\/265 + \(11\ x6\)\/265 \[LessEqual] 1, \(2\ x2\)\/139 + \(2\ x3\)\/139 + \(2\ x4\)\/139 + \(3\ \ x5\)\/139 \[LessEqual] 1, \(2\ x2\)\/139 + \(2\ x3\)\/139 + \(3\ x4\)\/139 + \(2\ \ x5\)\/139 \[LessEqual] 1, \(-\(\(4\ x1\)\/265\)\) + \(2\ x2\)\/265 + \(2\ x3\)\/265 + \ \(2\ x4\)\/265 + \(11\ x6\)\/265 \[LessEqual] 1, \(-\(\(2\ x1\)\/79\)\) + x2\/79 + \(4\ x6\)\/79 \[LessEqual] 1, \(-\(\(4\ x1\)\/265\)\) + \(2\ x2\)\/265 + \(2\ x3\)\/265 + \ \(2\ x5\)\/265 + \(11\ x6\)\/265 \[LessEqual] 1, \(3\ x2\)\/139 + \(2\ x3\)\/139 + \(2\ x4\)\/139 + \(2\ \ x5\)\/139 \[LessEqual] 1, x4\/36 + x6\/36 \[LessEqual] 1, \(2\ x3\)\/179 + \(2\ x4\)\/179 + \(2\ x5\)\/179 + \(5\ \ x6\)\/179 \[LessEqual] 1, \(-\(\(4\ x2\)\/265\)\) + \(2\ x3\)\/265 + \(2\ x4\)\/265 + \ \(11\ x5\)\/265 + \(2\ x6\)\/265 \[LessEqual] 1, \(-\(x2\/31\)\) + x4\/62 + \(3\ x5\)\/62 \[LessEqual] 1, \(-\(x2\/47\)\) + x4\/94 + \(2\ x5\)\/47 + x6\/94 \[LessEqual] 1, \(-\(\(2\ x2\)\/79\)\) + x4\/79 + \(4\ x5\)\/79 \[LessEqual] 1, x4\/31 + \(2\ x5\)\/93 \[LessEqual] 1, \(-\(x2\/50\)\) + \(3\ x4\)\/100 + \(3\ x5\)\/100 \[LessEqual] 1, \(3\ x3\)\/122 + \(3\ x5\)\/122 + x6\/61 \[LessEqual] 1, x3\/51 + x4\/51 + x5\/51 \[LessEqual] 1, \(-\(x2\/47\)\) + x3\/94 + x4\/94 + \(2\ x5\)\/47 \[LessEqual] 1, x5\/36 + x6\/36 \[LessEqual] 1, \(-\(\(4\ x2\)\/245\)\) + \(2\ x4\)\/245 + \(11\ x5\)\/245 + \ \(2\ x6\)\/245 \[LessEqual] 1, \(2\ x3\)\/109 + \(3\ x5\)\/109 + \(2\ x6\)\/109 \[LessEqual] 1, \(-\(\(4\ x2\)\/245\)\) + \(2\ x3\)\/245 + \(2\ x4\)\/245 + \ \(11\ x5\)\/245 \[LessEqual] 1, \(4\ x2\)\/655 + \(12\ x3\)\/655 + \(6\ x4\)\/655 + \(3\ \ x5\)\/131 + \(6\ x6\)\/655 \[LessEqual] 1, \(4\ x2\)\/565 + \(6\ x3\)\/565 + \(6\ x4\)\/565 + \(6\ \ x5\)\/565 + \(3\ x6\)\/113 \[LessEqual] 1, \(4\ x2\)\/307 + \(9\ x4\)\/307 + \(6\ x6\)\/307 \[LessEqual] 1, x4\/31 + \(2\ x6\)\/93 \[LessEqual] 1, \(2\ x3\)\/165 + x4\/33 + \(2\ x6\)\/165 \[LessEqual] 1, \(4\ x2\)\/397 + \(6\ x3\)\/397 + \(9\ x4\)\/397 + \(6\ \ x5\)\/397 \[LessEqual] 1, \(2\ x3\)\/165 + x4\/33 + \(2\ x5\)\/165 \[LessEqual] 1, x3\/36 + x6\/36 \[LessEqual] 1, \(3\ x3\)\/109 + \(2\ x5\)\/109 + \(2\ x6\)\/109 \[LessEqual] 1, x3\/38 + x4\/38 \[LessEqual] 1, \(2\ x4\)\/93 + x5\/31 \[LessEqual] 1, \(2\ x2\)\/265 - \(4\ x3\)\/265 + \(11\ x4\)\/265 + \(2\ \ x5\)\/265 + \(2\ x6\)\/265 \[LessEqual] 1, \(-\(\(4\ x3\)\/245\)\) + \(11\ x4\)\/245 + \(2\ x5\)\/245 + \ \(2\ x6\)\/245 \[LessEqual] 1, \(-\(\(2\ x3\)\/79\)\) + \(4\ x4\)\/79 + x5\/79 \[LessEqual] 1, \(-\(x3\/47\)\) + \(2\ x4\)\/47 + x5\/94 + x6\/94 \[LessEqual] 1, \(4\ x3\)\/307 + \(9\ x5\)\/307 + \(6\ x6\)\/307 \[LessEqual] 1, \(-\(\(4\ x2\)\/179\)\) + \(2\ x4\)\/179 + \(9\ x5\)\/179 \ \[LessEqual] 1, \(2\ x2\)\/109 + \(3\ x4\)\/109 + \(2\ x6\)\/109 \[LessEqual] 1, \(-\(\(4\ x3\)\/179\)\) + \(9\ x4\)\/179 + \(2\ x5\)\/179 \ \[LessEqual] 1, \(6\ x3\)\/307 + \(9\ x5\)\/307 + \(4\ x6\)\/307 \[LessEqual] 1, \(4\ x2\)\/629 + \(10\ x3\)\/629 + \(6\ x4\)\/629 + \(15\ x5\)\ \/629 + \(6\ x6\)\/629 \[LessEqual] 1, \(2\ x2\)\/131 + \(6\ x3\)\/655 + \(3\ x4\)\/131 + \(6\ \ x5\)\/655 + \(6\ x6\)\/655 \[LessEqual] 1, \(10\ x2\)\/629 + \(4\ x3\)\/629 + \(15\ x4\)\/629 + \(6\ x5\)\ \/629 + \(6\ x6\)\/629 \[LessEqual] 1, x4\/23 \[LessEqual] 1, \(6\ x2\)\/307 + \(9\ x4\)\/307 + \(4\ x6\)\/307 \[LessEqual] 1, \(2\ x2\)\/245 - \(4\ x3\)\/245 + \(11\ x4\)\/245 + \(2\ \ x5\)\/245 \[LessEqual] 1, \(9\ x3\)\/307 + \(4\ x5\)\/307 + \(6\ x6\)\/307 \[LessEqual] 1, x3\/33 + \(2\ x4\)\/165 + \(2\ x6\)\/165 \[LessEqual] 1, \(6\ x2\)\/655 + \(3\ x3\)\/131 + \(6\ x4\)\/655 + \(2\ \ x5\)\/131 + \(6\ x6\)\/655 \[LessEqual] 1, x3\/31 + \(2\ x6\)\/93 \[LessEqual] 1, \(2\ x2\)\/93 + x3\/31 \[LessEqual] 1, \(6\ x2\)\/397 + \(9\ x3\)\/397 + \(6\ x4\)\/397 + \(4\ \ x5\)\/397 \[LessEqual] 1, \(9\ x3\)\/307 + \(6\ x5\)\/307 + \(4\ x6\)\/307 \[LessEqual] 1, \(2\ x2\)\/265 + \(11\ x3\)\/265 - \(4\ x4\)\/265 + \(2\ \ x5\)\/265 + \(2\ x6\)\/265 \[LessEqual] 1, \(2\ x2\)\/245 + \(11\ x3\)\/245 - \(4\ x4\)\/245 + \(2\ \ x6\)\/245 \[LessEqual] 1, x3\/23 \[LessEqual] 1, \(2\ x2\)\/245 + \(11\ x3\)\/245 - \(4\ x4\)\/245 + \(2\ \ x5\)\/245 \[LessEqual] 1, \(2\ x2\)\/179 + \(9\ x3\)\/179 - \(4\ x4\)\/179 \[LessEqual] 1, \(2\ x2\)\/165 + x3\/33 + \(2\ x4\)\/165 \[LessEqual] 1, \(6\ x2\)\/629 + \(15\ x3\)\/629 + \(4\ x4\)\/629 + \(10\ x5\)\ \/629 + \(6\ x