Related papers: A rank computation problem related to the HK funct…

A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness…

Reduced modeling in high-dimensional reproducing kernel Hilbert spaces offers the opportunity to approximate efficiently non-linear dynamics. In this work, we devise an algorithm based on low rank constraint optimization and kernel-based…

In this paper, we are concerned with the computations of the $p$-rank of curves in two different setups. We first work with complete intersection varieties in $\mb{P}^n \text{ for}~n\ge 2$ and compute explicitly the action of Frobenius on…

Suppose B=F[x,y,z]/h is the homogeneous coordinate ring of a characteristic p degree 3 irreducible plane curve C with a node. Let J be a homogeneous (x,y,z)-primary ideal and n -> e_n be the Hilbert-Kunz function of B with respect to J. Let…

Low-rank tensor approximation techniques attempt to mitigate the overwhelming complexity of linear algebra tasks arising from high-dimensional applications. In this work, we study the low-rank approximability of solutions to linear systems…

Let $k[X] = k[x_{i,j}: i = 1,..., m; j = 1,..., n]$ be the polynomial ring in $m n$ variables $x_{i,j}$ over a field $k$ of arbitrary characteristic. Denote by $I_2(X)$ the ideal generated by the $2 \times 2$ minors of the generic $m \times…

We study maximal functions related to homogeneous polynomial hypersurfaces in $\mathbb{R}^3$. In a sense made precise in this paper, the region of $(p,q)$ for which we obtain $L^p\rightarrow L^q$ boundedness is optimal up to the endpoints…

This note is devoted to the study of the links between the Hilbert function of a subscheme X of the projective space, and its geometric properties. We will assume that X is arithmetically Cohen-Macaulay, which allows us to characterize its…

This paper introduces a directional multiscale algorithm for the two dimensional $N$-body problem of the Helmholtz kernel with applications to high frequency scattering. The algorithm follows the approach in [Engquist and Ying, SIAM Journal…

The Hilbert-Kunz multiplicity and $F$-signature are important invariants for researchers in commutative algebra and algebraic geometry. We provide software, and describe the automation of a calculation, for the two invariants in the case of…

The decompositions of an element of a finite von Neumann algebra into the sum of a normal operator plus an s.o.t.-quasinilpotent operator, obtained using the Haagerup--Schultz hyperinvariant projections, behave well with respect to…

Let X be a closed subscheme and let HF(X,-) and hp(X,-) denote, respectively, the Hilbert function and the Hilbert polynomial of X. We say that X has bipolynomial Hilbert function if HF(X,d)=min{hp(P^n,d),hp(X,d)} for every non-negative…

The Hilbert functions and the regularity of the graded components of local cohomology of a bigraded algebra are considered. Explicit bounds for these invariants are obtained for bigraded hypersurface rings.

In this article, we approach the Arnold corank problem, posed by Arnold in 1975, which asks whether the corank of holomorphic functions is an ambient topological invariant. Here, we obtain a complete positive answer to the metric Arnold…

We consider an off-diagonal self-adjoint finite rank perturbation of a self-adjoint operator in a complex separable Hilbert space $\mathfrak{H}_0 \oplus \mathfrak{H}_1$, where $\mathfrak{H}_1$ is finite dimensional. We describe the singular…

In the application of the Deift-Zhou steepest descent method to the Riemann-Hilbert problem for orthogonal polynomials, a model Riemann-Hilbert problem that appears in the multi-cut case is solved with the use of hyperelliptic theta…

In this paper we find an algorithm which computes the Hilbert function of schemes $Z$ of "fat points" in $\PP3$ whose support lies on a rational normal cubic curve $C$. The algorithm shows that the maximality of the Hilbert function in…

In the first part we survey some of the known results and conjectures on compact Hyperkaehler (HK) manifolds. In the second part we presents a program which aims to show that HK four-folds whose second cohomology (with 4-tuple cup-product)…

We study Hilbert functions, Lefschetz properties, and Jordan type of Artinian Gorenstein algebras associated to Perazzo hypersurfaces in projective space. The main focus lies on Perazzo threefolds, for which we prove that the Hilbert…

In the proof of the BFK-gluing formula for zeta-determinants of Laplacians there appears a real polynomial whose constant term is an important ingredient in the gluing formula. This polynomial is determined by geometric data on an…