Related papers: On three-variable expanders over finite valuation …

Let $\mathbb{F}_p$ be a prime field of order $p,$ and $A$ be a set in $\mathbb{F}_p$ with $|A| \leq p^{1/2}.$ In this note, we show that \[\max\{|A+A|, |f(A, A)|\}\gtrsim |A|^{\frac{6}{5}+\frac{4}{305}},\] where $f(x, y)$ is a…

Let $\mathbb{F}_q$ denote the finite field of $q$ elements with characteristic $p$. Let $\mathbb{Z}_q$ denote the unramified extension of the $p$-adic integers $\mathbb{Z}_p$ with residue field $\mathbb{F}_q$. In this paper, we investigate…

In this paper we prove new bounds for sums of convex or concave functions. Specifically, we prove that for all $A,B \subseteq \mathbb R$ finite sets, and for all $f,g$ convex or concave functions, we have $$|A + B|^{38}|f(A) + g(B)|^{38}…

We show that subsets of $\mathbb{R}^n$ of large enough Hausdorff and Fourier dimension contain polynomial patterns of the form \begin{align*} ( x ,\, x + A_1 y ,\, \dots,\, x + A_{k-1} y ,\, x + A_k y + Q(y) e_n ), \quad x \in…

We extend the proximity technique of Solymosi and Zahl [J. Combin. Theory, Ser. A (2024)] to the setting of trivariate polynomials. In particular, we prove the following result: Let $f(x,y,z)=(x-y)^2+(\varphi(x)-z)^2$, where $\varphi(x)\in…

Let $\F_q$ be a finite field of order $q$ and $P$ be a polynomial in $\F_q[x_1, x_2]$. For a set $A \subset \F_q$, define $P(A):=\{P(x_1, x_2) | x_i \in A \}$. Using certain constructions of expanders, we characterize all polynomials $P$…

Let $S \subset R$ be an arbitrary subset of a unique factorization domain $R$ and $\K$ be the field of fractions of $R$. The ring of integer-valued polynomials over $S$ is the set $\mathrm{Int}(S,R)= \{ f \in \mathbb{K}[x]: f(a) \in R\…

Let $f,g_1,\dots,g_m$ be polynomials with real coefficients in a vector of variables $x=(x_1,\dots,x_n)$. Denote by $\text{diag}(g)$ the diagonal matrix with coefficients $g=(g_1,\dots,g_m)$ and denote by $\nabla g$ the Jacobian of $g$. Let…

New and old results on closed polynomials, i.e., such polynomials f in K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n], are collected. Using some properties of closed polynomials we prove the following…

Let $P: \F \times \F \to \F$ be a polynomial of bounded degree over a finite field $\F$ of large characteristic. In this paper we establish the following dichotomy: either $P$ is a moderate asymmetric expander in the sense that $|P(A,B)|…

We study separable plus quadratic (SPQ) polynomials, i.e., polynomials that are the sum of univariate polynomials in different variables and a quadratic polynomial. Motivated by the fact that nonnegative separable and nonnegative quadratic…

A celebrated result by Hilbert says that every real nonnegative ternary quartic is a sum of three squares. We show more generally that every nonnegative quadratic form on a real projective variety $X$ of minimal degree is a sum of…

We will prove several expanders with exponent strictly greater than $2$. For any finite set $A \subset \mathbb R$, we prove the following six-variable expander results: \begin{align*} |(A-A)(A-A)(A-A)| &\gg…

Let $R$ be either the ring of Lipschitz quaternions, or the ring of Hurwitz quaternions. Then, $R$ is a subring of the division ring $\mathbb{D}$ of rational quaternions. For $S \subseteq R$, we study the collection $\rm{Int}(S,R) = \{f \in…

We establish several sum-product estimates over finite fields that involve polynomials and rational functions. First, |f(A)+f(A)|+|AA| is substantially larger than |A| for an arbitrary polynomial f over F_p. Second, a characterization is…

Gr\"obner bases have been generalized by replacing monomial orders with constructions such as valuations and filtrations. We consider suitable valuations on a rational valuation field $K(x,y)$ and analyze their behavior when restricting to…

The paper proves sum-of-square-of-rational-function based representations (shortly, sosrf-based representations) of polynomial matrices that are positive semidefinite on some special sets: $\mathbb{R}^n;$ $\mathbb{R}$ and its intervals…

We provide upper bounds on the total number of irreducible factors, and in particular irreducibility criteria for some classes of bivariate polynomials $f(x,y)$ over an arbitrary field $\mathbb{K}$. Our results rely on information on the…

Let $f\in \mathbb{R}[x_1,\ldots, x_k]$, for $k\ge 2$. For any finite sets $A_1,\ldots, A_k\subset \mathbb{R}$, consider the set $$ f(A_1,\ldots, A_k):=\{f(a_1,\ldots, a_k)\mid (a_1,\cdots,a_k)\in A_1\times\cdots \times A_k\}, $$ that is,…

A polynomial $f$ of degree $d$ and coefficients in an algebraically closed field $k$ defines a morphism $f:\mathbb{P}^1_k\longrightarrow\mathbb{P}^1_k$ which, if char$(k)\nmid d$, is unramified outside a finite set of points in the image:…