Multigraded Hilbert functions and toric complete intersection codes

Abstract

Let X be a complete n-dimensional simplicial toric variety with homogeneous coordinate ring S. We study the multigraded Hilbert function H Y of reduced 0-dimensional subschemes Y in X. We provide explicit formulas and prove non-decreasing and stabilization properties of H Y when Y is a 0-dimensional complete intersection in X. We apply our results to computing the dimension of some evaluation codes on 0-dimensional complete intersection in simplicial toric varieties.

Finite $0$-dimensional multiprojective schemes and their ideals

“…Remark 2.7. Part (1) of Proposition 2.6 is a generalization of Proposition 3.5 in [26] and Lemma 3.4 in [23].…”

“…, a k ) are then easily determined from Corollary 4.7 in [26]. In the literature, multigraded Hilbert functions of reduced 0-dimensional scheme on toric varieties can be found in [23] where the authors generalize some results on the Hilbert function in [26] and made some applications to Coding Theory. In this paper we are interested in generalizing some of the cited known results on the multigraded Hilbert function from sets of distinct points to 0-dimensional schemes in product of multiprojective spaces P n1 × • • • × P n k .…”

Numerical root finding via Cox rings

“…, P n ), is the coefficient of the monomial λ 1 λ 2 · · · λ n in Vol n ( n i=1 λ i P i ). A formula for the mixed volume that will be useful is (see [Bih16,ŞS16])…”

Multigraded Castelnuovo-Mumford regularity via Klyachko filtrations

“…Introduced in [17], Multigraded Castelnuovo-Mumford regularity of Cl(X)−graded R−modules has received a lot of attention in the last decades, particularly regarding to Cl(X)−graded ideals and their associated coordinate rings. See for instance [5,6,18,21].…”