Binomial Regular Sequences and Free Sums
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Binomial Regular Sequences and Free Sums Winfried Bruns Dedicated to Professor Ngo Viet Trung on the occasion of his sixtieth birthday
Received: 18 March 2014 / Revised: 10 May 2014 / Accepted: 16 May 2014 / Published online: 27 March 2015 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015
Abstract Recently, several authors have proved results on Ehrhart series of free sums of rational polytopes. In this note, we treat these results from an algebraic viewpoint. Instead of attacking combinatorial statements directly, we derive them from structural results on affine monoids and their algebras that allow conclusions for Hilbert and Ehrhart series. We characterize when a binomial regular sequence generates a prime ideal or even normality is preserved for the residue class ring. Keywords Polytope · Free sum · Ehrhart series Mathematics Subject Classification (2010) 52B20 · 13F20 · 14M25
1 Introduction Recently, several authors have proved results on Ehrhart series of free sums of rational polytopes (see Beck and Hos¸ten [1], Braun [5] and Beck et al. [2]). In this note, we treat these results from an algebraic viewpoint. Instead of attacking combinatorial statements directly, we derive them from structural results on affine monoids and their algebras that allow conclusions for Hilbert and Ehrhart series. This procedure follows the spirit of the monograph [6] to which the reader is referred for affine monoids and their algebras. Our approach is best explained by the motivating example, namely free sums of rational polytopes and their Ehrhartseries. The Ehrhart series of a rational polytope P is the ∞ k (formal) power series EP = k=0 E(P , k)t where E(P , k) counts the lattice points in the homothetic multiple kP ; see Beck and Robbins [3] for a gentle introduction to the fascinating area of Ehrhart series.
W. Bruns () Institut f¨ur Mathematik, Universit¨at Osnabr¨uck, 49069 Osnabr¨uck, Germany e-mail: [email protected]
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W. Bruns
One says that R = conv(P ∪ Q) is the free sum of the rational polytopes P and Q if 0 ∈ P ∩ Q, the vector subspaces RP and RQ intersect only in 0, and (Zm ∩ RR) = (Zm ∩ RP ) + (Zm ∩ RQ). It has been proved in [2, Theorem 1.4] that the Ehrhart series of the three polytopes are related by the equation ER = (1 − T )EP EQ
(*)
if and only if at least one of the polytopes P and Q is described by inequalities of type a1 x1 + · · · + am xm ≤ b with a1 , . . . , an ∈ Z and b ∈ {0, 1}. We approach the validity of (*) by considering the Ehrhart monoid E (P ) = (x, k) : x ∈ kP ∩ Zm = R+ (P × {1}) ∩ Zm+1 . The Ehrhart series is the Hilbert series of E (P ) or, equivalently, of the monoid algebra K[E (P )] over a field K, and therefore standard techniques for computing Hilbert series can be applied. Ehrhart monoids are normal: if nx ∈ E (P ) for some x in the group ZE (P ) and n ∈ Z+ , n > 0, then x ∈ E (P ). The normality of a monoid M is equivalent to the normality of K[M]. The free sum arises from the free join by a proj
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