Related papers: Large lower bounds for the betti numbers of graded…

Let $(A,\mathfrak{m})$ be a complete intersection ring of codimension $c\geq 2$ and dimension $d\geq 1$. Let $M$ be a finitely generated maximal Cohen-Macaulay $A$-module. Set $M_i=\text{Syz}^A_{i}(M)$. Let $e^{\mathfrak{m}}_i(M)$ be the…

We introduce and study the Bourbaki degree as a numerical invariant for \(2 \times 4\) matrices $\Theta$ of homogeneous polynomials over a polynomial ring \(R = k[x_1, \dots, x_n]\). This invariant, defined via a Bourbaki sequence for the…

Using constructions of the Whitham perturbation theory of integrable system we prove a new sharp upper bound of $3g/2-2$ on the dimension of complete subvarieties of $\M_g^{ct}$.

In this paper, we construct some piecewise defined functions, and study their $c$-differential uniformity. As a by-product, we improve upon several prior results. Further, we look at concatenations of functions with low differential…

In this article we extend a previous definition of Castelnuovo-Mumford regularity for modules over an algebra graded by a finitely generated abelian group. Our notion of regularity is based on Maclagan and Smith's definition, and is…

We initiate a classification of complex polynomials f of degree d having the top Betti number of the general fibre close to the maximum. We find a range in which the polynomial must have isolated singularities and another range where it may…

We describe the positive cone generated by bigraded Betti diagrams of artinian modules of codimension two, whose resolutions become pure of a given type when taking total degrees. If the differences of these total degrees, p and q, are…

Let $K$ be a field and let $S = K[X_1, \ldots, X_n]$. Let $I$ be a graded ideal in $S$ and let $M$ be a finitely generated graded $S$-module. We give upper bounds on the regularity of Koszul homology modules $H_i(I, M)$ for several classes…

A minimal geodesic on a Riemannian manifold is a geodesic defined on $\mathbb{R}$ that lifts to a globally distance minimizing curve on the universal covering. Bangert proved that there is a lower bound for the number of geometrically…

The bigraded Betti numbers b^{-i,2j}(P) of a simple polytope P are the dimensions of the bigraded components of the Tor groups of the face ring k[P]. The numbers b^{-i,2j}(P) reflect the combinatorial structure of P as well as the topology…

Let $K$ be an algebraically closed field of characteristic $0$ and $G$ a finite abelian group. For a $G$-graded $K$-algebra $A$, we define the primeness property for graded central polynomials: for any graded polynomials $f$ and $g$ in…

We give a formula to compute all the top degree graded Betti numbers of the path ideal of a cycle. Also we will find a criterion to determine when Betti numbers of this ideal are non zero and give a formula to compute its projective…

In this paper, we prove the degree upper bound of projective subschemes in terms of the reduction number and show that the maximal cases are only arithmetically Cohen-Macaulay subschemes with linear resolution. Furthermore, it can be shown…

The border rank of the matrix multiplication operator for n by n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n^2-n. Our bounds…

For a graph $G$, let $f_2(G)$ denote the largest number of vertices in a $2$-regular subgraph of $G$. We determine the minimum of $f_2(G)$ over $3$-regular $n$-vertex simple graphs $G$. To do this, we prove that every $3$-regular multigraph…

Let $R$ be a polynomial ring over a field and $M= \bigoplus_n M_n$ a finitely generated graded $R$-module, minimally generated by homogeneous elements of degree zero with a graded $R$-minimal free resolution $\mathbf{F}$. A Cohen-Macaulay…

We prove a new lower bound for the Mahler measure of a polynomial in one and in several variables that depends on the complex coefficients, and the number of monomials. In one variable our result generalizes a classical inequality of…

We investigate the structure of polynomials of degree four in many variables over a fixed prime field $\mathbb{F}=\mathbb{F}_{p}$. In 2007, Green and Tao proved that if a polynomial $f:\mathbb{F}^{n}\rightarrow\mathbb{F}$ is poorly…

Given an integer $\Delta \ge 3$, let ${\cal G}_{\Delta }$ be the set of connected graphs $G\neq K_{\Delta +1}$ with maximum degree $\Delta $ and, for $i=1,\cdots, \Delta $, let $V_i(G)$ be the set of vertices of $G$ of degree $i$. \\ We…

The $\beta$-generalized quasi-geostrophic equation is studied in the range of $\alpha \in (0, 1), \beta \in (1/2, 1), 1/2 < \alpha + \beta < 3/2$. When $\alpha \in (1/2, 1), \beta \in (1/2, 1)$ such that $1 \leq \alpha + \beta < 3/2$, using…