Periods of Coefficients of Ehrhart

Quasipolynomials of 2-Dimensional

PolytopesBy Christopher O’Neill and Anastasia Chavez

Topics

•Review of Preliminary Topics / Problem

•Proof of Formula used in Program

•Overview of Pari

•Overview of the Code

•Final Remaining Steps

Review of Topic

•Definition(s) of a Polytope and equivalence

•Lattice point counting

•Dilates of Polytopes

•Ehrhart Quasipolynomial

•Ehrhart Series (new)

Computing the Ehrhart Quasipolynomial

•g(z)/h(z) = sum(t >= 0 of f(t)*z^t)

•[g(z), h(z) polynomials, deg(g) < deg(h), all roots of h(z) are pth roots of unity, at least one them has multiplicity d+1] => f(t) is a quasipolynomial

Overview of Pari

•Command-line algebraic simplification

•Pari vs. Maple

•Time

•Proper simplification

•Piping through stdin

Overview of Pari

•Notes on use:• Reading GPRC: /etc/gprc ...Done.

GP/PARI CALCULATOR Version 2.1.6 (released)...parisize = 4000000, primelimit = 500000

? (t+1)*(t+2)%1 = t^2 + 3*t + 2? \qGood bye!

•Don’t want all this garbage, use -q option

Overview of Pari

•With -q option:? (t+1)*(t+2)t^2 + 3*t + 2? \q

•Much better, but still colored: use -test

•Final output:? (t+1)*(t+2)t^2 + 3*t + 2? \q

Overview of Pari

•temp.lat.rat from Latte:(((((((1) + (-1)*t^1) * (((1) + (-1)*t^1) * (-183)))^1) * (412)) + ((-1) * ((((1) + (-1)*t^1) * ((((1) + (-1)*t^1) * (-16653)) + ((191) * (-183)))) + ((229) * (((1) + (-1)*t^1) * (-183)))) * (1))) / ((((1) + (-1)*t^1) * (((1) + (-1)*t^1) * (-183)))^2)) + ((-1) * (((((((1) + (-1)*t^1) * ((-183) * ((1) + (-1)*t^1)))^1) * (229)) + ((-1) * ((((1) + (-1)*t^1) * (((-183) * (8)) + ((-16653) * ((1) + (-1)*t^1)))) + ((46) * ((-183) * ((1) + (-1)*t^1)))) * (1))) / ((((1) + (-1)*t^1) * ((-183) * ((1) + (-1)*t^1)))^2)))

•Output from ‘gp -q -test < temp.lat.rat’:(-t - 1)/(t^3 - 3*t^2 + 3*t - 1)

•Much easier to deal with!

Overview of Code

•Coded in C, using gcc

•Split into parts:

•Generate random 2D Polytope

•Create its Vertex file for CDD

Overview of Code

•Split into parts (continued):

•Run CDD, convert file to Latte format

•Run Ehrhart, simplify output with Pari

•Extract quasipolynomial from series

•Extract periods of Coefficients

Overview of Code•Generating 2D polytope:

•User supplies # vertices, min/max values

•Generate random vertices (x1/x2, y1/y2), where min <= x1, x2, y1, y2 <= max

•Store in structure and return

•Find denominator as maximal period

Overview of Code

•Can also specify a specific 2D polytope

•Our example: (0,0), (1,2/3), (3,0)

(0,0(0,0))

(3,0(3,0))

(1,2/3(1,2/3))

Overview of Code

•Create vertex file for CDD by outputting the vertices into a file in the correct format

•Sample file:V-representationbegin3 3 rational1 0/1 0/11 1/1 2/31 3/1 0/1end

•Run CDD to get hyperplane file

Overview of Code

•Output file from CDD:* cdd+: Double Description Method in C++:Version 0.77dev1 (Jan. 25, 2000)* Copyright (C) 1999, Komei Fukuda, fukuda@ifor.math.ethz.ch* Compiled for Rational Exact Arithmetic with GMP...*Total processor time = 0 seconds* = 0h 0m 0s*Since hull computation is chosen, the output is a minimal inequality system*FINAL RESULT:*Number of Facets = 3H-representationbegin3 3 rational 3 -1 -3 0 0 1 0 1 -3/2end

Overview of Code

•Convert CDD file for use with Latte

•Remove garbage

•Convert to integers

•Converted file:3 3 3 -1 -3 0 0 1 0 2 -3

•Run ehrhart command, get unsimplified file

Overview of Code

•Unsimplified file:x := (t^1 / ((((1) + (-1)*t^1) * (((1) + (-1)*t^1) * ((1) + (-1)*t^1)))^1)) + ((-1) * (t^2 / ((((1) + (-1)*t^1) * (((1) + (-1)*t^3) * ((1) + (-1)*t^1)))^1))) + ((-1) * (((((((1) + (-1)*t^1) * (((1) + (-1)*t^1) * (-152)))^1) * (152)) + ((-1) * ((((1) + (-1)*t^1) * ((((1) + (-1)*t^1) * (-11476)) + ((-238)*t^1 * (-152)))) + ((-1185)*t^1 * (((1) + (-1)*t^1) * (-152)))) * (1))) / ((((1) + (-1)*t^1) * (((1) + (-1)*t^1) * (-152)))^2))) + (t^2 / ((((1) + (-1)*t^1) * (((1) + (-1)*t^3) * ((1) + (-1)*t^1)))^1)) + ((((((-152) * (((1) + (-1)*t^1) * ((1) + (-1)*t^1)))^1) * (0)) + ((-1) * (((-152) * ((((1) + (-1)*t^1) * (-881)*t^1) + ((-390)*t^1 * ((1) + (-1)*t^1)))) + ((-11476) * (((1) + (-1)*t^1) * ((1) + (-1)*t^1)))) * (1))) / (((-152) * (((1) + (-1)*t^1) * ((1) + (-1)*t^1)))^2)):

•Modify file for Pari:(t^1 / ((((1) + (-1)*t^1) * (((1) + (-1)*t^1) * ((1) + (-1)*t^1)))^1)) + ((-1) * (t^2 / ((((1) + (-1)*t^1) * (((1) + (-1)*t^3) * ((1) + (-1)*t^1)))^1))) + ((-1) * (((((((1) + (-1)*t^1) * (((1) + (-1)*t^1) * (-152)))^1) * (152)) + ((-1) * ((((1) + (-1)*t^1) * ((((1) + (-1)*t^1) * (-11476)) + ((-238)*t^1 * (-152)))) + ((-1185)*t^1 * (((1) + (-1)*t^1) * (-152)))) * (1))) / ((((1) + (-1)*t^1) * (((1) + (-1)*t^1) * (-152)))^2))) + (t^2 / ((((1) + (-1)*t^1) * (((1) + (-1)*t^3) * ((1) + (-1)*t^1)))^1)) + ((((((-152) * (((1) + (-1)*t^1) * ((1) + (-1)*t^1)))^1) * (0)) + ((-1) * (((-152) * ((((1) + (-1)*t^1) * (-881)*t^1) + ((-390)*t^1 * ((1) + (-1)*t^1)))) + ((-11476) * (((1) + (-1)*t^1) * ((1) + (-1)*t^1)))) * (1))) / (((-152) * (((1) + (-1)*t^1) * ((1) + (-1)*t^1)))^2))

Overview of Code

•Simplified file from Pari:(t + 1)/(-t^3 + 3*t^2 - 3*t + 1)

•Expanded g(z) polynomial:t^7 + 4*t^6 + 9*t^5 + 13*t^4 + 13*t^3 + 9*t^2 + 4*t + 1

•Quasipolynomial constituents1/1*t^2 2/1*t^1 1/1*t^0 1/1*t^2 2/1*t^1 1/1*t^0 1/1*t^2 2/1*t^1 1/1*t^0

•Lack of pretty print only temporary!

Overview of Code

•Find periods of each coefficient

•Starting at 1, loop through divisors of denominator

•Stop when we find a valid period

•C code given on next slidevoid getPeriods(Polynomial *subPolys, int *p0, int *p1, int period)

Overview of Code• per0 = 1;

while(per0 < period) { matchFound = 1; for(j = 0;j < per0;j++) { initial = subPolys[j].coefficients[2]; frac = subPolys[j].fractions[2]; for(k = 0;k < period / per0;k++) { if (subPolys[(k * per0) + j].coefficients[2] != initial || subPolys[(k * per0) + j].fractions[2] != frac) { matchFound = 0; } } } if (matchFound) { break; } per0++; while(period % per0 != 0 && per0 < period) per0++; }

Overview of Code•For our example, both periods are 1,

since all 3 constituents of the quasipolynomial are the same polynomial

•Worst-case running time: O(d^2), where d is the denominator of the polytope

•Final line of output from program:NumVerts: 3, Verts: (0/1, 0/1), (1/1, 2/3), (3/1, 0/1), Periods: p0 = 1, p1 = 1, denom = 3...

Possible Improvements

•Pretty print

•Simplifying fractions

•Eliminate unnecessary vertices (CDD)

•Saving data to a database rather than printing a lot of garbage to cmd line

•Add cmd line options (verbosity, quiet, etc.)

Expected Results

•Next steps:

•Run program a massive # of times to produce a ton of random polytopes

•Check data for period collapse, look for period combinations that don’t appear