Related papers: Schubert determinantal ideals are Hilbertian

One important question in algebraic complexity is understanding the complexity of polynomial ideals (Grochow, Bulletin of EATCS 131, 2020). Andrews and Forbes (STOC 2022) studied the determinantal ideals $I^{\det}_{n,m,r}$ generated by the…

In this paper, we present a modular strategy which describes key properties of the absolute primary decomposition of an equidimensional polynomial ideal defined by polynomials with rational coefficients. The algorithm we design is based on…

We study when blowup algebras are $F$-split or strongly $F$-regular. Our main focus is on algebras given by symbolic and ordinary powers of ideals of minors of a generic matrix, a symmetric matrix, and a Hankel matrix. We also study ideals…

Let S be the Cox ring of a product of r projective spaces. In this paper, we study the Cartwright-Sturmfels Hilbert schemes of S, which are multigraded Hilbert schemes that only parametrize radical ideals. Our main result shows that these…

The paper investigates the behavior of Hilbert-Samuel and Hilbert-Kunz multiplicities in families of ideals. It is shown that Hilbert-Samuel multiplicity is upper semicontinuous almost generally and that Hilbert-Kunz multiplicity is upper…

An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in…

We prove that an inclusion-exclusion inspired expression of Schubert polynomials of permutations that avoid the patterns 1432 and 1423 is nonnegative. Our theorem implies a partial affirmative answer to a recent conjecture of Yibo Gao about…

The Hilbert function of a module over a positively graded algebra is of quasi-polynomial type (Hilbert--Serre). We derive an upper bound for its grade, i.e. the index from which on its coefficients are constant. As an application, we give a…

Let C be a uniform clutter, i.e., all the edges of C have the same size, and let A be the incidence matrix of C. We denote the column vectors of A by v1,...,vq. The vertex covering number of C, denoted by g, is the smallest number of…

In this PhD thesis we propose an algorithmic approach to the study of the Hilbert scheme. Developing algorithmic methods, we also obtain general results about Hilbert schemes. In Chapter 1 we discuss the equations defining the Hilbert…

We show that any nonzero polynomial in the ideal generated by the $r \times r$ minors of an $n \times n$ matrix $X$ can be used to efficiently approximate the determinant. For any nonzero polynomial $f$ in this ideal, we construct a small…

In this paper we introduce $p-$Ferrer diagram, note that $1-$ Ferrer diagram are the usual Ferrer diagrams or Ferrer board, and corresponds to planar partitions. To any $p-$Ferrer diagram we associate a $p-$Ferrer ideal. We prove that…

Conjectures for the Hilbert function of the m-th symbolic power of the ideal of n general points of P2 are verified for infinitely many m for each square n > 9, using an approach developed by the authors in a previous paper. In those cases…

The Gr\"obner stratum of a monomial ideal $\id{j}$ is an affine variety that parametrizes the family of all ideals having $\id{j}$ as initial ideal (with respect to a fixed term ordering). The Gr\"obner strata can be equipped in a natural…

Let $X$ be a matrix with entries in a polynomial ring over an algebraically closed field $K$. We prove that, if the entries of $X$ outside some $(t \times t)$-submatrix are algebraically dependent over $K$, the arithmetical rank of the…

We give a bound on the Castelnuovo-Mumford regularity of a homogeneous ideals I, in a polynomial ring A, in terms of number of variables and the degrees of generators, when the dimension of A/I is at most two. This bound improves the one…

For a partition $\lambda$ of $n \in {\mathbb N}$, let $I^{\rm Sp}_\lambda$ be the ideal of $R=K[x_1,\ldots,x_n]$ generated by all Specht polynomials of shape $\lambda$. In the previous paper, the second author showed that if $R/I^{\rm…

In this paper we study some combinatorial properties of the shuffle algebra $S_{q_{1},q_{2}}$ with two complex parameters and, in particular, obtain the Hilbert series for $S_{q_{1},q_{2}}/I$, where $I$ is the ideal generated by $z-\lambda$…

We present a new technique for computing Hilbert series of N=1 supersymmetric QCD in four dimensions with unitary and special unitary gauge groups. We show that the Hilbert series of this theory can be written in terms of determinants of…

The main focus of this paper is on the problem of relating an ideal $I$ in the polynomial ring $\mathbb Q[x_1, \dots, x_n]$ to a corresponding ideal in $\mathbb F_p[x_1,\dots, x_n]$ where $p$ is a prime number; in other words, the…