Let I be a homogeneous Artinian ideal in a polynomial ring R = k[x1, . . . , xn] over a field k of characteristic 0. We study an equivalent condition for the generic initial ideal gin(I) with respect to reverse lexicographic order to be almost reverse lexicographic. As a result, we show that Moreno-Socias conjecture implies Fröberg conjecture. And for the case codimI ≤ 3, we show that R/I has t...

Let A = K[x1, . . . , xn] denote the polynomial ring in n variables over a field K of characteristic 0 with each deg xi = 1. Given arbitrary integers i and j with 2 ≤ i ≤ n and 3 ≤ j ≤ n, we will construct a monomial ideal I ⊂ A such that (i) βk(I) < βk(Gin(I)) for all k < i, (ii) βi(I) = βi(Gin(I)), (iii) βl(Gin(I)) < βl(Lex(I)) for all l < j and (iv) βj(Gin(I)) = βj(Lex(I)), where Gin(I) is t...

Abstract We consider the Allen–Cahn equation $${\partial _t}{u}-\Delta u=u-u^3$$ ∂ t u - Δ = 3 with a rapidly mixing Gaussian field as init...

Fixed any term-ordering < on the set T(n) of terms in n variables x1, . . . , xn, we study homogeneous ideal a ⊆ P(n) of the ring of polynomials in n variables over a field k, via the associated order-ideal N (a), consisting of all the terms which are not maximal terms of elements of a and called sous-éscalier of a [7, 12, 14]. In particular we will focus our attention on the following combinat...

Let K be a field, S a polynomial ring and E an exterior algebra over K, both in a finite set of variables. In this paper we study the graded Betti numbers of graded ideals in S and E. First, we prove that if the graded Betti number β ii+k(S/I) = β S ii+k(S/Gin(I)) for some i > 1 and k ≥ 0 then one has β qq+k(S/I) = β S qq+k(S/Gin(I)) for all q ≥ i, where I ⊂ S is a graded ideal. Second, we show...