1,721,207 research outputs found
A NOTE ON THE V-INVARIANT
No full textLet R be a finitely generated N-graded algebra domain over a Noetherian ring and let I be a homogeneous ideal of R. Given P ∈ Ass(R/I) one defines the v-invariant vP(I) of I at P as the least c ∈ N such that P = I : f for some f ∈ Rc. A classical result of Brodmann [Proc. Amer. Math. Soc. 74 (1979), pp. 16–18] asserts that Ass(R/In) is constant for large n. So it makes sense to consider a prime ideal P ∈ Ass(R/In) for all the large n and investigate how vP(In) depends on n. We prove that vP(In) is eventually a linear function of n. When R is the polynomial ring over a field this statement has been proved independently also by Ficarra and Sgroi in their recent preprint [Asymptotic behaviour of the v-number of homogeneous ideals, https://arxiv.org/abs/2306.14243, 2023]
SAGBI combinatorics of maximal minors and a SAGBI algorithm
No full textThe maximal minors of a matrix of indeterminates are a universal Grobner basis by a theorem of Bernstein, Sturmfels and Zelevinsky. On the other hand it is known that they are not always a universal SAGBI basis. By an experimental approach we discuss their behavior under varying monomial orders and their extensions to SAGBI bases. These experiments motivated a new implementation of the SAGBI algorithm which is organized in a Singular script and falls back on Normaliz for the combinatorial computations. In comparison to packages in the current standard distributions of Macaulay 2, version 1.21, and Singular, version 4.2.1 and a package intended for CoCoA 5.4.2, it extends the range of computability by at least one order of magnitude.& COPY; 2023 Elsevier Ltd. All rights reserved
Lovász-Saks-Schrijver ideals and coordinate sections of determinantal varieties
No full textMotivated by questions in algebra and combinatorics we study two ideals associated to a simple graph G: • the Lovász-Saks-Schrijver ideal defining the d-dimensional orthogonal representations of the graph complementary to G, and • the determinantal ideal of the (d+1)-minors of a generic symmetric matrix with 0 in positions prescribed by the graph G. In characteristic 0 these two ideals turn out to be closely related and algebraic properties such as being radical, prime or a complete intersection transfer from the Lovász-Saks-Schrijver ideal to the determinantal ideal. For Lovász-Saks-Schrijver ideals we link these properties to combinatorial properties of G and show that they always hold for d large enough. For specific classes of graphs, such a forests, we can give a complete picture and classify the radical, prime and complete intersection Lovász-Saks- Schrijver ideals
Regularity of primes associated with polynomial parametrisations
No full textWe prove a doubly exponential bound for the Castelnuovo-Mumford
regularity of prime ideals defining varieties with polynomial parametrisation
Radical Generic Initial Ideals
No full textIn this paper, we survey the theory of Cartwright–Sturmfels ideals. These are Zn-graded ideals, whose multigraded generic initial ideal is radical. Cartwright–Sturmfels ideals have surprising properties, mostly stemming from the fact that their Hilbert scheme only contains one Borel-fixed point. This has consequences, e.g., on their universal Gröbner bases and on the family of their initial ideals. In this paper, we discuss several known classes of Cartwright–Sturmfels ideals and we find a new one. Among determinantal ideals of same-size minors of a matrix of variables and Schubert determinantal ideals, we are able to characterize those that are Cartwright–Sturmfels
Invariant rings of the special orthogonal group have nonunimodal h -vectors
No full textFor K an infinite field of characteristic other than two, consider the action of the special orthogonal group SOt(K) on a polynomial ring via copies of the regular representation. When K has characteristic zero, Boutot's theorem implies that the invariant ring has rational singularities; when K has positive characteristic, the invariant ring is F-regular, as proven by Hashimoto using good filtrations. We give a new proof of this, viewing the invariant ring for SOt(K) as a cyclic cover of the invariant ring for the corresponding orthogonal group; this point of view has a number of useful consequences, for example, it readily yields the a-invariant and information on the Hilbert series. Indeed, we use this to show that the h-vector of the invariant ring for SOt(K) need not be unimodal
Sagbi bases, defining ideals and algebras of minors
No full textThis paper extends the article of Bruns and Conca on Sagbi bases and their computation (J. Symb. Comput. 120 (2024)) in two directions. (i) We describe the extension of the Singular library sagbiNormaliz.sing to the computation of defining ideals of subalgebras of polynomial rings. (ii) We give a complete classification of the algebras of minors for which the generating set is a Sagbi basis with respect to a suitable monomial order and we identify universal Sagbi basis in three cases. The investigation is illustrated by several examples
Multidegrees, prime ideals, and non-standard gradings
No full textWe study several properties of multihomogeneous prime ideals. We show that the multigraded generic initial ideal of a prime has very special properties, for instance, its radical is Cohen-Macaulay. We develop a comprehensive study of multidegrees in arbitrary positive multigraded settings. In these environments, we extend the notion of Cartwright-Sturmfels ideals by means of a standardization technique. Furthermore, we recover or extend important results in the literature, for instance: we provide a multidegree version of Hartshorne’s result stating the upper semicontinuity of arithmetic degree under flat degenerations, and we give an alternative proof of Brion’s result regarding multiplicity-free varieties
IDEALS GENERATED BY POWER SUMS
No full textWe consider ideals in a polynomial ring generated by collections of power sum polynomials, and obtain conditions under which these define complete intersection rings, normal domains, and unique factorization domains. We also settle a key case of a conjecture of Conca, Krattenthaler, and Watanabe, and prove other results in that direction
Castelnuovo-Mumford Regularity and Powers
No full textThis note has two goals. The first is to give a short and self contained introduction to the Castelnuovo-Mumford regularity for standard graded rings R= iEN Ri over general base rings R0. The second is to present a simple and concise proof of a classical result due to Cutkosky, Herzog and Trung and, independently, to Kodiyalam asserting that the regularity of powers Iv of an homogeneous ideal I of R is eventually a linear function in v. Finally we show how the flexibility of the definition of the Castelnuovo-Mumford regularity over general base rings can be used to give a simple proof of a result proved by the authors in “Maximal minors and linear powers”
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