of 26/26
Periods of Degree 2 Ehrhart Quasipolynomials Christopher O’Neill and Anastasia Chavez

# Periods of Degree 2 Ehrhart Quasipolynomials Christopher O’Neill and Anastasia Chavez

• View
217

0

Embed Size (px)

### Text of Periods of Degree 2 Ehrhart Quasipolynomials Christopher O’Neill and Anastasia Chavez

• Periods of Degree 2 Ehrhart QuasipolynomialsChristopher ONeill and Anastasia Chavez

• TopicsReview of polytopes, lattice point countingResults from readingsLatte introComplete Tour de Latte via an easy exampleOther examples in LattePlan of attack for remainder of project

• Review of TopicV-Representations / H-RepresentationsInteger point enumerationEhrhart quasipolynomialDenominator of polytope / period of quasipolynomialPeriod collapse

• Results from ReadingsMcAllister - Period collapse can happen with arbitrarily large denominatorsMethod of proof: cut into pieces, rearrangeExample of rational polytope whose polynomial is not the polynomial of any integral polytopeConjectures - Equidecomposable

• Results from ReadingsMcAllister, Woods - Can find a polytope and quasipolynomial given dimension, denominator, and period2 Dimensions: Period collapses iff Lp(t) = At^2 + (B/2)t + 1, where A is the area of P, and B is the number of boundary points

• LatteLATTice point EnumerationCounts lattice points, generates Ehrhart seriesWe will use it to find Ehrhart quasipolynomials of random 2d polytopes

• LatteAll command line based3 main commands:cdd - convert between H, V repscount - count lattice pointsehrhart - find Ehrhart series

• LatteNote - count/ehrhart only allow H-repsDenominator => need V-repThus, we need cdd

• Latte ExampleRight triangle in 2 spaceV-representation - (0,0), (0,5), (5,0)H-representation - x1 >= 0, x2 >= 0, x1 + x2
• Latte ExampleFirst step: cdd to convert V- to H-repInput file triangle.ext, outputs triangle.ineFormat of triangle.ext - * comments like this * m = # of vertices * d = dimension of polytope * numbertype = integral / rational V-representation begin m d+1 numbertype 1 v1 v2 ... ... ... end

• Latte ExampleOur triangle.ext - V-representation begin 3 3 integer 1 0 0 1 5 0 1 0 5 endExecute cdd triangle.ext

• Latte Examplecdd outputs triangle.ine - * cdd+: Double Description Method in C++:Version 0.77dev1 (Jan. 25, 2000) * Copyright (C) 1999, Komei Fukuda, [email protected] * Compiled for Rational Exact Arithmetic with GMP *Input File:triangle.ext(3x3) *HyperplaneOrder: LexMin *Degeneracy preknowledge for computation: None (possible degeneracy) *Hull computation is chosen. *Computation starts at Tue Mar 27 14:35:38 2007 * terminates at Tue Mar 27 14:35:38 2007 *Total processor time = 0 seconds * = 0h 0m 0s *Since hull computation is chosen, the output is a minimal inequality system *FINAL RESULT:-representation begin 3 3 rational 5 -1 -1 0 1 0 0 0 1 end

• Latte ExampleSecond step - count lattice pointsInput file triangle.lat, output to promptFormat of triangle.lat - * comments like this are actually not allowed * pretend they are not here * m = # of hyperplanes (not necessarily # of facets!) * d = dimension of polytope * integers only! no fractions like cdd m d+1 b1 -a11 -a12 ... b2 -a21 -a22 ... ...

• Latte ExampleOur triangle.lat - 3 3 5 -1 -1 0 1 0 0 0 1 10 0 1Note the lack of commentsAlso note the extra hyperplane we added

• Latte ExampleCommand line output excerpts: Checking whether the input polytope is empty or not... Removing redundant inequalities and finding hidden equalities.... ... Ax
• Latte ExampleExtra Latte function: ehrhartFinds Ehrhart series, not Ehrhart polynomialSame input file as count, outputs triangle.lat.rat, with unsimplified series

• Latte ExampleOutput file triangle.lat.rat - x := (((((((1) + (-1)*t^1) * ((-933) * (-155)))^2) * (591328)) + ((-1) * ((((1) + (-1)*t^1) * (((-933) * (-11935)) + ((-434778) * (-155)))) + ((-1028)*t^1 * ((-933) * (-155)))) * ((((((1) + (-1)*t^1) * ((-933) * (-155)))^1) * (1088)) + ((-1) * ((((1) + (-1)*t^1) * (((-933) * (-11935)) + ((-434778) * (-155)))) + ((-1028)*t^1 * ((-933) * (-155)))) * (1)))) + ((-1) * (((1) + (-1)*t^1) * ((-933) * (-155))) * ((((1) + (-1)*t^1) * (((-933) * (-608685)) + ((-434778) * (-11935)) + ((-134926106) * (-155)))) + ((-1028)*t^1 * (((-933) * (-11935)) + ((-434778) * (-155)))) + ((-527878)*t^1 * ((-933) * (-155)))) * (1))) / ((((1) + (-1)*t^1) * ((-933) * (-155)))^3)) + ((((((-933) * (((1) + (-1)*t^1) * (-778)))^2) * (6612066)) + ((-1) * (((-933) * ((((1) + (-1)*t^1) * (-302253)) + ((3637) * (-778)))) + ((-434778) * (((1) + (-1)*t^1) * (-778)))) * (((((-933) * (((1) + (-1)*t^1) * (-778)))^1) * (3637)) + ((-1) * (((-933) * ((((1) + (-1)*t^1) * (-302253)) + ((3637) * (-778)))) + ((-434778) * (((1) + (-1)*t^1) * (-778)))) * (1)))) + ((-1) * ((-933) * (((1) + (-1)*t^1) * (-778))) * (((-933) * ((((1) + (-1)*t^1) * (-78182776)) + ((3637) * (-302253)) + ((6612066) * (-778)))) + ((-434778) * ((((1) + (-1)*t^1) * (-302253)) + ((3637) * (-778)))) + ((-134926106) * (((1) + (-1)*t^1) * (-778)))) * (1))) / (((-933) * (((1) + (-1)*t^1) * (-778)))^3)) + ((-1) * ((((((-155) * ((-778) * ((1) + (-1)*t^1)))^2) * (302253)) + ((-1) * (((-155) * (((-778) * (-253)*t^1) + ((-302253) * ((1) + (-1)*t^1)))) + ((-11935) * ((-778) * ((1) + (-1)*t^1)))) * (((((-155) * ((-778) * ((1) + (-1)*t^1)))^1) * (778)) + ((-1) * (((-155) * (((-778) * (-253)*t^1) + ((-302253) * ((1) + (-1)*t^1)))) + ((-11935) * ((-778) * ((1) + (-1)*t^1)))) * (1)))) + ((-1) * ((-155) * ((-778) * ((1) + (-1)*t^1))) * (((-155) * (((-778) * (-31878)*t^1) + ((-302253) * (-253)*t^1) + ((-78182776) * ((1) + (-1)*t^1)))) + ((-11935) * (((-778) * (-253)*t^1) + ((-302253) * ((1) + (-1)*t^1)))) + ((-608685) * ((-778) * ((1) + (-1)*t^1)))) * (1))) / (((-155) * ((-778) * ((1) + (-1)*t^1)))^3))):Needs to be simplified!!!

• Latte ExampleMaple - algebraic manipulation software, similar to MathematicaAfter input to Maple, it gives us the series (1 + 18t + 6t^2)/(1-t)^3

• More Latte ExamplesRational Polytopes - Another triangleVertices - (0,0), (0, 3), (1, 2/3)Hyperplanes - x2 >= 0, x1 + x2
• More Latte Examplesrational.ext - V-representation begin 3 3 rational 1 0 0 1 1 2/3 1 3 0 endOutput of cdd rational.ext - H-representation begin 3 3 rational 3 -1 -3 0 0 1 0 1 -3/2 end

• More Latte Examplesrational.lat (modified rational.ine) - 3 3 3 -1 -3 0 0 1 0 2 -3Some Output from count rational.lat - Checking whether the input polytope is empty or not... Removing redundant inequalities and finding hidden equalities.... Ax
• Harder Latte Example7th dilate of the 5-cubeVertices - {0,7}^5Hyperplanes - 0
• Harder Latte Examplelarge.ext - V-representation begin 32 6 integral 1 0 0 0 0 0 1 7 0 0 0 0 1 0 7 0 0 0 ... endlarge.ine - H-representation begin 10 6 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 ... end

• Harder Latte ExampleOutput from count large.lat Ax
• Next Steps to TakeWrite an algorithm to generate random rational 2-polytopes, and determine their Ehrhart quasipolynomial and periodImplement this algorithm using Latte commands and either Perl or C++Use the implementation to generate lots of examples of rational 2-polytopes and their Ehrhart quasipolynomials / periods

• ReferencesTyrrell B. McAllister. Quasi-period Collapse in Rational Polytopes. 2007. Tyrrell B. McAllister, Kevin M. Woods. The Minimul Period of the Ehrhart Quasipolynomial of a Rational Polytope. 2007. Mattias Beck, Sinai Robins. Computing the Continuous Discretely. Springer, 2006.

Documents
Education
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Automotive
Documents
Documents
Documents
Documents