periods of degree 2 ehrhart quasipolynomials christopher o’neill and anastasia chavez
TRANSCRIPT
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Periods of Degree 2 Ehrhart
QuasipolynomialsChristopher O’Neill and Anastasia Chavez
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Topics•Review of polytopes, lattice point
counting
•Results from readings
•Latte intro
•Complete Tour de Latte via an easy example
•Other examples in Latte
•Plan of attack for remainder of project
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Review of Topic
•V-Representations / H-Representations
•Integer point enumeration
•Ehrhart quasipolynomial
•Denominator of polytope / period of quasipolynomial
•Period collapse
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Results from Readings
•McAllister - Period collapse can happen with arbitrarily large denominators
•Method of proof: cut into pieces, rearrange
•Example of rational polytope whose polynomial is not the polynomial of any integral polytope
•Conjectures - Equidecomposable
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Results from Readings
•McAllister, Woods - Can find a polytope and quasipolynomial given dimension, denominator, and period
•2 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
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Latte
•LATTice point Enumeration
•Counts lattice points, generates Ehrhart series
•We will use it to find Ehrhart quasipolynomials of random 2d polytopes
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Latte
•All command line based
•3 main commands:
•cdd - convert between H, V rep’s
•count - count lattice points
•ehrhart - find Ehrhart series
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Latte
•Note - count/ehrhart only allow H-rep’s
•Denominator => need V-rep
•Thus, we need cdd
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Latte Example
•Right triangle in 2 space
•V-representation - (0,0), (0,5), (5,0)
•H-representation - x1 >= 0, x2 >= 0, x1 + x2 <= 5
•# of lattice points - 21
•Ehrhart polynomial - (25/2)t^2 + (15/2)t + 1
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Latte Example
•First step: cdd to convert V- to H-rep
•Input file triangle.ext, outputs triangle.ine
•Format of triangle.ext - * comments like this* m = # of vertices* d = dimension of polytope* numbertype = integral / rationalV-representationbeginm d+1 numbertype1 v1 v2 .........end
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Latte Example
•Our triangle.ext - V-representationbegin3 3 integer1 0 01 5 01 0 5end
•Execute “cdd triangle.ext”
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Latte Example
•cdd 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:-representationbegin3 3 rational 5 -1 -1 0 1 0 0 0 1end
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Latte Example
•Second step - count lattice points
•Input file triangle.lat, output to prompt
•Format 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 cddm d+1b1 -a11 -a12 ...b2 -a21 -a22 ......
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Latte Example
•Our triangle.lat - 3 35 -1 -10 1 00 0 110 0 1
•Note the lack of comments
•Also note the extra hyperplane we added
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Latte Example
•Command line output excerpts:Checking whether the input polytope is empty or not...Removing redundant inequalities and finding hidden equalities....
...Ax <= b, given as (b|-A):=========================[5 -1 -1][0 1 0][0 0 1]
Time: 0.004 sec
...***** Total number of lattice points: 21 ****
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Latte Example
•Extra Latte function: “ehrhart”
•Finds Ehrhart series, not Ehrhart polynomial
•Same input file as “count,” outputs triangle.lat.rat, with unsimplified series
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Latte Example•Output 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!!!
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Latte Example
•Maple - algebraic manipulation software, similar to Mathematica
•After input to Maple, it gives us the series (1 + 18t + 6t^2)/(1-t)^3
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More Latte Examples
•Rational Polytopes - Another triangle
•Vertices - (0,0), (0, 3), (1, 2/3)
•Hyperplanes - x2 >= 0, x1 + x2 <= 3, 3x2 - 2x1 <= 0
•Lattice Point count - 4
•Ehrhart Polynomial - t^2 + 2t + 1
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More Latte Examples
•rational.ext - V-representationbegin3 3 rational1 0 01 1 2/31 3 0end
•Output of “cdd rational.ext” - H-representationbegin3 3 rational3 -1 -30 0 10 1 -3/2end
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More Latte Examples
•rational.lat (modified rational.ine) - 3 33 -1 -30 0 10 2 -3
•Some Output from “count rational.lat” - Checking whether the input polytope is empty or not...Removing redundant inequalities and finding hidden equalities....
Ax <= b, given as (b|-A):=========================[3 -1 -3][0 0 1][0 2 -3]
***** Total number of lattice points: 4 ****Time: 0.016 sec
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Harder Latte Example
•7th dilate of the 5-cube
•Vertices - {0,7}^5
•Hyperplanes - 0 <= xi <= 7
•Count - (t+1)^d = 8^5 = 32768
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Harder Latte Example
• large.ext - V-representationbegin32 6 integral1 0 0 0 0 01 7 0 0 0 01 0 7 0 0 0...end
• large.ine - H-representationbegin10 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
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Harder Latte Example
•Output from “count large.lat”Ax <= b, given as (b|-A):=========================[0 0 0 0 0 1][0 1 0 0 0 0][0 0 1 0 0 0][0 0 0 1 0 0][0 0 0 0 1 0][7 0 0 0 0 -1][7 0 0 0 -1 0][7 0 0 -1 0 0][7 0 -1 0 0 0][7 -1 0 0 0 0]...***** Total number of lattice points: 32768 ****Computation done. Time: 0.048003 sec
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Next Steps to Take
•Write an algorithm to generate random rational 2-polytopes, and determine their Ehrhart quasipolynomial and period
• Implement 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
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References•Tyrrell 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.