Related papers: Maximum linearizations of lower sets in $\mathbb{N…

Consider real bivariate polynomials f and g, respectively having 3 and m monomial terms. We prove that for all m>=3, there are systems of the form (f,g) having exactly 2m-1 roots in the positive quadrant. Even examples with m=4 having 7…

Let $I$ be a homogeneous ideal in a polynomial ring over a field. Let $I^{(n)}$ be the $n$-th symbolic power of $I$. Motivated by results about ordinary powers of $I$, we study the asymptotic behavior of the regularity function $\text{reg}~…

Let $ R $ be a regular local ring with maximal ideal $ \mathfrak{m} $. We consider elements $ f \in R $ such that their Newton polyhedron has a loose edge. We show that if the symbolic restriction of $f$ to such an edge is a product of two…

Let $\mathbb{I}$ denote an imaginary quadratic field or the field $\mathbb{Q}$ of rational numbers and $\mathbb{Z}_{\mathbb{I}}$ its ring of intergers. We shall prove an explicit Baker type lower bound for $\mathbb{Z}_{\mathbb{I}}$-linear…

We determine upper bounds for the maximum order of an element of a finite almost simple group with socle T in terms of the minimum index m(T) of a maximal subgroup of T: for T not an alternating group we prove that, with finitely many…

In this thesis we are interested in describing some homological invariants of certain classes of monomial ideals. We will pay attention to the squarefree and non-squarefree lexsegment ideals.

Let $(A, \mathfrak m)$ be a normal two-dimensional local ring and $I$ an $\mathfrak m$-primary integrally closed ideal with a minimal reduction $Q$. Then we calculate the numbers: $\mathrm{nr}(I) = \min\{n \;|\; \overline{I^{n+1}} =…

We introduce the combinatorial Lyubeznik resolution of monomial ideals. We prove that this resolution is isomorphic to the usual Lyubezbnik resolution. As an application, we give a combinatorial method to determine if an ideal is a…

Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal generated in degree $d$. Bandari and Herzog conjectured that a monomial ideal $I$ is polymatroidal if and only if all its monomial…

We consider the semiconjugate factorization and reduction of order for non-autonomous, nonlinear, higher order difference equations containing linear arguments. These equations have appeared in several mathematical models in biology and…

For the set of graphs with a given degree sequence, consisting of any number of $2's$ and $1's$, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of $m$-matchings. We find the…

Young flattenings, introduced by Landsberg and Ottaviani, give determinantal equations for secant varieties and their non-vanishing provides lower bounds for border ranks of tensors and in particular polynomials. We study monomial-optimal…

Let $\mathcal O$ be a holomorphy ring in a global field $K$, and $R$ a classical maximal $\mathcal O$-order in a central simple algebra over $K$. We study sets of lengths of factorizations of cancellative elements of $R$ into atoms…

We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\Pi_{2^{m}}^*$ represent the set of polynomials computable with…

Let $J\subset S=K[x_0,...,x_n]$ be a monomial strongly stable ideal. The collection $\Mf(J)$ of the homogeneous polynomial ideals $I$, such that the monomials outside $J$ form a $K$-vector basis of $S/I$, is called a {\em $J$-marked…

Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization problem. This degree is related to the number of…

In this paper, we give a lower bound for the maximum size of a $k$-colored sum-free set in $\mathbb{Z}_m^n$, where $k\geq 3$ and $m\geq 2$ are fixed and $n$ tends to infinity. If $m$ is a prime power, this lower bound matches (up to lower…

We introduce the notion of Bernstein-Sato polynomial of an arbitrary variety (which is not necessarily reduced nor irreducible), using the theory of V-filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing filtration by…

For a given subset $A\subseteq \mathbb F_q^*$, we study the problem of finding a large packing set $B$ of $A$, that is, a set $B \subseteq \mathbb F_q^*$ such that $|AB|=|A||B|$. We prove the existence of such a $B$ of size $|B|\ge…

For a family of sets $\mathcal{F}$, let $\omega(\mathcal{F}):=\sum_{\{A,B\}\subset \mathcal{F}}|A\cap B|$. In this paper, we prove that provided $n$ is sufficiently large, for any $\mathcal{F}\subset \binom{[n]}{k}$ with $|\mathcal{F}|=m$,…