Related papers: Remarks on polynomial count varieties

Let $X$ be an algebraic variety over a finite field $\bF_q$, homogeneous under a linear algebraic group. We show that the number of rational points of $X$ over $\bF_{q^n}$ is a periodic polynomial function of $q^n$ with integer…

We reprove the results of Jordan [18] and Siebert [31] and show that the Lie algebra of polynomial vector fields on an irreducible affine variety X is simple if and only if X is a smooth variety. Given proof is self-contained and does not…

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more…

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…

In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for…

We study the problem of counting the total number of affine solutions of a system of n binomials in n variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is…

Let $k$ be any field and $k^s$ its separable closure. Let $X$ be an affine variety over $k$ which is isomorphic to affine $n$-space over the field extension $k^s$. Then $X$ is isomorphic to affine $n$ space over $k$.

We use tools of additive combinatorics for the study of subvarieties defined by {\it high rank} families of polynomials in high dimensional $\mathbb{F} _q$-vector spaces. In the first, analytic part of the paper we prove a number properties…

An algebraic variety $X$ is called a homogeneous variety if the automorphism group $\mathrm{Aut}(X)$ acts on $X$ transitively, and a homogeneous space if there exists a transitive action of an algebraic group on $X$. We prove a criterion of…

Most results on the value sets $V_f$ of polynomials $f \in \mathbb{F}_q[x]$ relate the cardinality $|V_f|$ to the degree of $f$. In particular, the structure of the spectrum of the class of polynomials of a fixed degree $d$ is rather well…

We investigate the relation between the Hodge theory of a smooth subcanonical $n$-dimensional projective variety $X$ and the deformation theory of the affine cone $A_X$ over $X$. We start by identifying $H^{n-1,1}_{\mathrm{prim}}(X)$ as a…

In the area of symbolic-numerical computation within computer algebra, an interesting question is how "close" a random input is to the "critical" ones, like the singular matrices in linear algebra or the polynomials with multiple roots for…

We give a counting formula in terms of modified Hall-Littlewood polynomials and the chromatic quasisymmetric function for the number of points on an arbitrary Hessenberg variety over a finite field. As a consequence, we express the…

We introduce a new matroid (graph) invariant, the arboricity polynomial. Given a matroid, the arboricity polynomial enumerates the number of covers of the ground set by disjoint independent sets. We establish the polynomiality of the…

Given a complex algebraic variety X, we define a natural number called the motivic dimension which measures the amount of transcendental (co)homology of X. It is zero precisely when all the (co)homolgy is spanned by algebraic cycles. Most…

Trinomial varieties are affine varieties given by a system of equations consisting of polynomials with three terms. Such varieties are total coordinate spaces of normal varieties with torus action of complexity one. For an affine variety…

We study over a number field, the iterates of automorphisms of the affine space. More precisely, we are interested in the periodic and non-periodic points; for the former the questions are similar to the ones about torsion points on abelian…

We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. We investigate basic properties of the polynomial invariants including stability…

We give several new criteria for a quasi-projective variety to be affine. In particular, we prove that an algebraic manifold $Y$ with dimension $n$ is affine if and only if $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and $\kappa(D,…

Let $\mathcal{F}_n$ be the set of unitary polynomials of degree $n \ge 2$ that have their roots in $\mathbb{Z}^*$. We note $$ Q(x) := x^n+a_{1}x^{n-1}+\dots+a_{n}. $$ We show that any two fixed consecutive coefficients $(a_{j},a_{j+1})$ ($j…