Related papers: A sharper estimate on the Betti numbers of sets de…

We determine the density of integral binary forms of given degree that have squarefree discriminant, proving for the first time that the lower density is positive. Furthermore, we determine the density of integral binary forms that cut out…

We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme and…

A numerical semigroup is an additive subsemigroup of the non-negative integers. In this paper, we consider parametrized families of numerical semigroups of the form $P_n = \langle f_1(n), \ldots, f_k(n) \rangle$ for polynomial functions…

Let $M$ be a finitely generated module over a local ring $(R,\mathfrak{m})$. By $\mathcal{S}_j(M)$, we denote the $j$th symmetric power of $M$ ($j$th graded component of the symmetric algebra $\mathcal{S}_R(M)$). The purpose of this paper…

Given a polynomial $f$ and a semi-algebraic set $S$, we provide a symbolic algorithm to find the equations and inequalities defining a semi-algebraic set $Q$ which is identical to the closure of the image of $S$ under $f$, i.e.,…

For a polynomial $g(x)$ of deg $k \geq 2$ with integer coefficients and positive integer leading coefficient, we prove an upper bound for the least prime $p$ such that $g(p)$ is in non-homogeneous Beatty sequence $\lbrace \lfloor \alpha…

We give new positive and negative results (some conditional) on speeding up computational algebraic geometry over the reals: (1) A new and sharper upper bound on the number of connected components of a semialgebraic set. Our bound is novel…

In this thesis we investigate certain types of monomial ideals of polynomial rings over fields. We are interested in minimal free resolutions of these ideals (or equivalently the quotients of the polynomial ring by the ideals) considered as…

We give a bound for the Betti numbers of the Stanley-Reisner ring of a stellar subdivision of a Gorenstein* simplicial complex by applying unprojection theory. From this we derive a bound for the Betti numbers of iterated stellar…

In this paper, the Boiti Leon Pempinelli system in (2+1)-dimensions is revisited for Lie symmetries and invariant solutions. An infinite-dimensional Lie algebra is obtained using the Lie invariance criterion and is further classified into…

Let B_{2t} be a bipartite planar graph with an even number of regions. We are able to find bounds for the graded Betti numbers and the projective dimension of the quotient ring associated to the graph. We also will investigate the minimal…

A classical theorem of Baranyai states that, given integers $2\leq k < n$ such that $k$ divides $n$, one can find a family of ${n-1\choose k-1}$ partitions of $[n]$ into $k$-element subsets such that every subset appears in exactly one…

Let $k \ge 2$ and $\alpha_1, \beta_1, ..., \alpha_k, \beta_k$ be reals such that the $\alpha_i$'s are irrational and greater than 1. Suppose further that some ratio $\alpha_i/\alpha_j$ is irrational. We study the representations of an…

Let $\mathrm{R}$ be a real closed field and $Q_1, \ldots, Q_{\ell} \in \mathrm{R}[X_1, \ldots,X_k]$ such that for each $i, 1 \leq i \leq \ell$, $\mathrm{deg} (Q_i) \leq d_i$. For $1 \leq i \leq \ell$, denote by $\mathcal{Q}_i = \{Q_1,…

Let $T$ be a perfect binary tree and $I$ be its edge ideal in the polynomial ring $S$. We determine the vertex cover number, independent number, and establish the recursive formula to compute the number of minimal vertex covers. As a…

We prove a smooth analogue of the classical Thom-Milnor bound, showing that the Betti numbers of the zero set of a smooth map on a compact Riemannian manifold can be controlled by a condition number computed from its first jet. This extends…

We derive new estimates for the first Betti number of compact Riemannian manifolds. Our approach relies on the Birman-Schwinger principle and Schatten norm estimates for semigroup differences. In contrast to previous works we do not require…

In this article bipartite planar graphs St_r are investigated, r the number of their plane regions. Bounds for the graded Betti numbers and the projective dimension of the quotient ring associated to such graphs are discussed. We prove that…

We consider the open problem of determining the graded Betti numbers for fat point subschemes supported at general points of the projective plane. We relate this problem to the open geometric problem of determining the splitting type of the…

We consider the problem of approximating the reachable set of a discrete-time polynomial system from a semialgebraic set of initial conditions under general semialgebraic set constraints. Assuming inclusion in a given simple set like a box…