Related papers: A sharp degree bound in the real Jacobian conjectu…

Let $k$ be a field of characteristic $p > 0$ such that $[k:k^p] < \infty$ and let $f \in R = k[x_0, ..., x_n]$ be homogeneous of degree $d$. We obtain a sharp bound on the degrees in which the Frobenius action on $H^n_\mathfrak{m}(R/fR)$…

We classify all post-critically finite unicritical polynomials defined over the maximal totally real algebraic extension of ${\mathbb Q}$. Two auxiliary results used in the proof of this result may be of some independent interest. The first…

The main result of this paper is to prove some type of Real Jacobian Conjecture. It is proved by the Minimax Principle and asserts if the eigenvalues of $F'(x)$ are bounded from zero and all the eigenvalues of $F'(x)+F'(x)^T$ are strictly…

We show that an integer-valued quadratic polynomial on $\mathbb{R}^2$ can not be injective on the integer lattice points of any affine convex cone if its discriminant is nonzero. A consequence is the non-existence of quadratic packing…

Fix integers $r\geq 4$ and $i\geq 2$. Let $C$ be a non-degenerate, reduced and irreducible complex projective curve in $\mathbb P^r$, of degree $d$, not contained in a hypersurface of degree $\leq i$. Let $p_a(C)$ be the arithmetic genus of…

Using Stickelberger's theorem on Gauss sums, we show that if $F$ is a planar function on a finite field $\mathbb{F}_q$, then for all non-zero functions $G : \mathbb{F}_q \to \mathbb{F}_q$, we have \begin{equation*} d_{\mathsf{alg}}(G \circ…

Let $I$ be a squarefree monomial ideal of a polynomial algebra over a field minimally generated by $f_1,...,f_r$ of degree $ d\geq 1$, and a set $E$ of monomials of degree $\geq d+1$. Let $J\subsetneq I$ be a squarefree monomial ideal…

Based on the reduction of degree in polynomial mappings and some known results in algebraic geometry, by introducing the Brouwer degree, a tool from differential topology, algebraic topology and algebraic geometry, we completely prove the…

We prove a conjecture of Bonamy, Bousquet, Pilipczuk, Rz\k{a}\.zewski, Thomass\'e, and Walczak, that for every graph $H$, there is a polynomial $p$ such that for every positive integer $s$, every graph of average degree at least $p(s)$…

D. Dimitrov has posed the problem of finding polynomials that set the sharpness of the Koebe Quarter Theorem for polynomials and asked whether Suffridge polynomials are optimal. We disprove Dimitrov's conjecture for polynomials of degree 3,…

We introduce the concept of injective category number $\text{IC}(f)$ for a continuous map $f\colon X\to~Y$, and present fundamental results concerning this numerical invariant. The value $\text{IC}(f)$ quantifies the \aspas{complexity} or…

Over an algebraically closed field $\mathbb{F}$ of zero characteristic polynomial map $\xi: \mathbb{F}^n\rightarrow \mathbb{F}^n$ of the form $\xi(x)=x-((xA_1)^{3}, (xA_2)^{3},..., (xA_n)^{3})$, where $x=(x_1,x_2,...,x_n)$ a row vector of…

We give necessary and sufficient conditions in terms of sign vectors for the injectivity of families of polynomial maps with arbitrary real exponents defined on the positive orthant. Our work relates and extends existing injectivity…

Let $l$ be a finite field of cardinality $q$ and let $n$ be in $\mathbb{Z}_{\geq 1}$. Let $f_1,\ldots,f_n \in l[x_1,\ldots,x_n]$ not all constant and consider the evaluation map $f=(f_1,\ldots,f_n) \colon l^n \to l^n$. Set…

Let $k$ be a fixed integer. We determine the complexity of finding a $p$-partition $(V_1, \dots, V_p)$ of the vertex set of a given digraph such that the maximum out-degree of each of the digraphs induced by $V_i$, ($1\leq i\leq p$) is at…

We prove that a polynomial map is invertible if and only if some associated differential ring homomorphism is bijective. To this end, we use a theorem of Crespo and Hajto linking the invertibility of polynomial maps with Picard-Vessiot…

We show that if $G$ is a connected graph of maximum degree at most $4$, which is not $C_{2,5}$, then the strong matching number of $G$ is at least $\frac{1}{9}n(G)$. This bound is tight and the proof implies a polynomial time algorithm to…

In this paper, we first show that the Jacobian Conjecture is true for non-homogeneous power linear mappings under some conditions. Secondly, we prove an equivalent statement about the Jacobian Conjecture in dimension $r\geq 1$ and give some…

In this article we analyze the global diffeomorphism property of polynomial maps $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$ by studying the properties of the Newton polytopes at infinity corresponding to the sum of squares polynomials…

Let $\Sigma(f)$ be critical points of a polynomial $f \in \mathbb{K}[x,y]$ in the plane $\mathbb{K}^2$, where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$. Our goal is to study the critical point map $\mathfrak{S}_d$, by sending polynomials…