Period collapse in Ehrhart quasi-polynomials Tyrrell B. McAllister 1 Joint work with ene Rochais 1 H´ el` 1 University of Wyoming University of Kansas 22 May 2016 T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 1 / 22
Introduction — Polytopes: convex and non-convex Definition A convex rational polytope Q ⊆ R n is the convex hull of a finite subset S ⊆ Q n . If S ⊆ Z n , then Q is integral . A rational polytope Q ⊆ R n (not necessarily convex) is a topological ball that is a union � Q = Q i i ∈ I for some finite family {Q i : i ∈ I } of convex rational polytopes, all with the same a ffi ne span. T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 2 / 22
Introduction — Counting integer points Fix a rational polytope Q ⊆ R n . Consider positive integer dilates: 4P 2P 1P We are interested in the function counting the number of integer lattice points in the k th dilate: k �→ | k Q ∩ Z n | . T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 3 / 22
Introduction — Examples Let Q := [0 , 1] ⊆ R . Then | k Q ∩ Z | = k + 1. T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 4 / 22
Introduction — Examples Let Q := [0 , 1] ⊆ R . Then | k Q ∩ Z | = k + 1. Let Q := [0 , 1 p ] ⊆ R for p ∈ Z ≥ 1 . Then � k � � k � � k � + 1 = 1 + 1 = 1 | k Q ∩ Z | = pk − pk + , p p p � � � � � � k is the fractional part of k k k where p , and = 1 − . p p p T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 4 / 22
Introduction — Examples Let Q := [0 , 1] ⊆ R . Then | k Q ∩ Z | = k + 1. Let Q := [0 , 1 p ] ⊆ R for p ∈ Z ≥ 1 . Then � k � � k � � k � + 1 = 1 + 1 = 1 | k Q ∩ Z | = pk − pk + , p p p � � � � � � k is the fractional part of k k k where p , and = 1 − . p p p Let Q := [0 , 1] × [0 , 1 p ] ⊆ R 2 . Then � 1 � k �� � � k Q ∩ Z 2 � � = ( k + 1) pk + p � 1 � k �� � k � = 1 p k 2 + p + k + . p p T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 4 / 22
Introduction — Examples Let Q := [0 , 1] ⊆ R . Then | k Q ∩ Z | = k + 1. Let Q := [0 , 1 p ] ⊆ R for p ∈ Z ≥ 1 . Then � k � � k � � k � + 1 = 1 + 1 = 1 | k Q ∩ Z | = pk − pk + , p p p � � � � � � k is the fractional part of k k k where p , and = 1 − . p p p Let Q := [0 , 1] × [0 , 1 p ] ⊆ R 2 . Then � 1 � k �� � � k Q ∩ Z 2 � � = ( k + 1) pk + p � 1 � k �� � k � = 1 p k 2 + p + k + . p p � � k Note: k �→ is a periodic function Z → Q with period p . p T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 4 / 22
Introduction — Ehrhart’s theorem Theorem (Ehrhart 1962) Let ehr Q ( k ) := | k Q ∩ Z n | for k ∈ Z ≥ 1 . Then ehr Q ( k ) is a rational quasi-polynomial function of k. That is, 2 ( k ) k 2 + · · · + c Q ehr Q ( k ) = c Q 0 ( k ) + c Q 1 ( k ) k + c Q n ( k ) k n for some periodic functions c Q i : Z → Q . T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 5 / 22
Introduction — Ehrhart’s theorem Theorem (Ehrhart 1962) Let ehr Q ( k ) := | k Q ∩ Z n | for k ∈ Z ≥ 1 . Then ehr Q ( k ) is a rational quasi-polynomial function of k. That is, 2 ( k ) k 2 + · · · + c Q ehr Q ( k ) = c Q 0 ( k ) + c Q 1 ( k ) k + c Q n ( k ) k n for some periodic functions c Q i : Z → Q . Theorem (Ehrhart 1962 — Integral case) Moreover, if Q is integral, then ehr Q ( k ) is a polynomial function of k: ehr Q ( k ) = c Q 0 + c Q 1 k + · · · + c Q n k n . That is, all the c Q i ’s are constant functions. T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 5 / 22
Introduction — Our motivating question Definition In general, let � Q be the ring of periodic functions Z → Q . Then � Q [ x ] is the ring of quasi-polynomials . (Note: � Q [ x ] contains Q [ x ] as a subring.) Definition Let Q ⊆ R n be a rational polytope. Call ehr Q ( x ) ∈ � Q [ x ] the Ehrhart quasi-polynomial of Q (or Ehrhart polynomial if ehr Q ( x ) ∈ Q [ x ]). Motivating Question Which quasi-polynomials are the Ehrhart quasi-polynomials of rational polytopes? That is, when does f ( x ) ∈ � Q [ x ] satisfy f ( x ) = ehr Q ( x ) for some rational polytope Q ? T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 6 / 22
Introduction — The polynomial case . . . Many necessary conditions on Ehrhart polynomials of integral polytopes are known. Conditions on their coe ffi cients: Stanley (1980, 1991); Betke & McMullen (1985); Hibi (1994), Haase & Nill & Payne (2009); Henk & Tagami (2009); Stapledon (2009). Conditions on their roots in C : Beck & De Loera & Develin & Pfeifle & Stanley (2005); Braun (2008); Pfeifle (2010). T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 7 / 22
Introduction — The polynomial case . . . Many necessary conditions on Ehrhart polynomials of integral polytopes are known. Conditions on their coe ffi cients: Stanley (1980, 1991); Betke & McMullen (1985); Hibi (1994), Haase & Nill & Payne (2009); Henk & Tagami (2009); Stapledon (2009). Conditions on their roots in C : Beck & De Loera & Develin & Pfeifle & Stanley (2005); Braun (2008); Pfeifle (2010). Many nontrivial results! But still cannot characterize the Ehrhart polynomials of: convex integral polytopes in dimension > 2 (!), convex rational polytopes ∗ in dimension > 1 (!!). ∗ That is, pseudo-integral polytopes : rational polytopes Q such that ehr Q ( x ) is a polynomial. T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 7 / 22
Introduction — Recalibrating our question Original Motivating Question (restated) Which are the possible periodic functions c i : Z → Q that appear in n � c i ( x ) x i , ehr Q ( x ) = i =0 for rational polytopes Q ⊆ R n ? Humbled by the “merely” polynomial case, we adjust our goals: New Motivating Question Which are the possible periods of the functions c i : Z → Q that appear in n � c i ( x ) x i , ehr Q ( x ) = i =0 for rational polytopes Q ⊆ R n ? T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 8 / 22
Introduction — Period sequences Definition Fix a rational polytope Q . Let p i ∈ Z ≥ 1 be the period of the coe ffi cient c Q in ehr Q ( x ). The period sequence of Q is the tuple ( p 0 , . . . , p n ). i Fact: If Q ⊆ R n has non-empty interior, then the “leading coe ffi cient” c Q n is a constant. Indeed, c Q n = Vol( Q ). Hence, p n = 1. New Motivating Question (re-restated) Which tuples ( p 0 , . . . , p n − 1 , 1) ∈ ( Z ≥ 1 ) n are period sequences of rational polytopes? T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 9 / 22
Introduction — Period sequences Definition Fix a rational polytope Q . Let p i ∈ Z ≥ 1 be the period of the coe ffi cient c Q in ehr Q ( x ). The period sequence of Q is the tuple ( p 0 , . . . , p n ). i Fact: If Q ⊆ R n has non-empty interior, then the “leading coe ffi cient” c Q n is a constant. Indeed, c Q n = Vol( Q ). Hence, p n = 1. New Motivating Question (re-restated) Which tuples ( p 0 , . . . , p n − 1 , 1) ∈ ( Z ≥ 1 ) n are period sequences of rational polytopes? Answer In not-necessarily-convex case: All such tuples. In convex case: Open — but at least all of the form ( p 0 , p 1 , 1 , . . . , 1). T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 9 / 22
Main Results Theorem (TM & H. Rochais, 2016+) There exists a not-necessarily-convex rational polytope Q ⊆ R n with period sequence ( p 0 , . . . , p n − 1 , 1) , for all p 0 , . . . , p n − 1 ∈ Z ≥ 1 . Theorem (TM & H. Rochais, 2016+) There exists a convex rational polytope Q ⊆ R n with period sequence ( p 0 , p 1 , 1 , . . . , 1) , for all p 0 , p 1 ∈ Z ≥ 1 . T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 10 / 22
McMullen’s bound on coe ffi cient periods — (Preliminaries) � � � � ( − 1 3 , 1 3 ) , ( 1 3 , 2 ( − 2 3 , 1 2 ) , ( 2 3 , 1 Conv 3 ) Conv 2 ) Definition A polytope is reticular i ff its a ffi ne span contains a lattice point. Reticular Not reticular For every dimension i ≤ n , can ask: “Is every i -face reticular?” T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 11 / 22
McMullen’s bound on coe ffi cient periods — (Preliminaries) � � � � ( − 1 3 , 1 3 ) , ( 1 3 , 2 ( − 2 3 , 1 2 ) , ( 2 3 , 1 Conv 3 ) Conv 2 ) Definition A polytope is reticular i ff its a ffi ne span contains a lattice point. Reticular Not reticular For every dimension i ≤ n , can ask: “Is every i -face reticular?” Definition Fix polytope Q ⊆ R n . For 0 ≤ i ≤ n , the i th McMullen index of Q is m i := min { k ∈ Z ≥ 1 : every i -face of kP is reticular } . Example Polytope on left has m 0 = 3 and m 1 = 1. Polytope on right has m 0 = 6 and m 1 = 2. T. B. McAllister (U. of Wyoming) Ehrhart quasi-polynomials KU, 22 May 2016 11 / 22
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