Related papers: Rational Solutions of First Order Algebraic Ordina…

Let $f(t, y,y')=\sum_{i=0}^d a_i(t, y)y'^i=0$ be a first order ordinary differential equation with polynomial coefficients. Eremenko in 1999 proved that there exists a constant $C$ such that every rational solution of $f(t, y,y')=0$ is of…

For R(z, w) rational with complex coefficients, of degree at least 2 in w, we show that the number of rational functions f(z) solving the difference equation f(z+1)=R(z, f(z)) is finite and bounded just in terms of the degrees of R in the…

We prove that $\mathrm{deg}(f) \leq \widetilde{O}(\mathrm{rdeg}(f)^3)$ for every Boolean function $f$, where $\mathrm{deg}(f)$ is the degree of $f$ and $\mathrm{rdeg}(f)$ is the rational degree of $f$. This resolves the second of the three…

We consider algebraic ordinary differential equations (AODEs) and study their polynomial and rational solutions. A sufficient condition for an AODE to have a degree bound for its polynomial solutions is presented. An AODE satisfying this…

Let $F(x, y)$ be a binary form with integer coefficients, degree $n\geq 3$ and irreducible over the rationals. Suppose that only $s + 1$ of the $n + 1$ coefficients of $F$ are nonzero. We show that the Thue inequality $|F(x,y)|\leq m$ has…

We propose an algebraic geometric approach for studying rational solutions of first-order algebraic ordinary difference equations. For an autonomous first-order algebraic ordinary difference equations, we give an upper bound for the degrees…

An astonishing fact was established by Lee A. Rubel (1981): there exists a fixed non-trivial fourth-order polynomial differential algebraic equation (DAE) such that for any positive continuous function $\varphi$ on the reals, and for any…

The approximate non-deterministic degree of a Boolean function $f$, denoted $\mathsf{ndeg}_\epsilon(f)$ (written $\mathsf{N}_\epsilon(f)$ for brevity), is the minimum degree of a real polynomial $p$ such that $0 \le |p(x)| \le \epsilon$…

We consider the differential equations y''=\lambda_0(x)y'+s_0(x)y, where \lambda_0(x), s_0(x) are C^{\infty}-functions. We prove (i) if the differential equation, has a polynomial solution of degree n >0, then \delta_n=\lambda_n…

In this paper we characterize real bivariate polynomials which have a small range over large Cartesian products. We show that for every constant-degree bivariate real polynomial $f$, either $|f(A,B)|=\Omega(n^{4/3})$, for every pair of…

In this paper, it is shown that if F(x , y) is an irreducible binary form with integral coefficients and degree $n \geq 3$, then provided that the absolute value of the discriminant of F is large enough, the equation |F(x , y)| = 1 has at…

Let $F\in\mathbb{C}[x,y,s,t]$ be an irreducible constant-degree polynomial, and let $A,B,C,D\subset\mathbb{C}$ be finite sets of size $n$. We show that $F$ vanishes on at most $O(n^{8/3})$ points of the Cartesian product $A\times B\times…

Let $\mathbf{f} = (f_1, \ldots, f_R)$ be a system of polynomials with integer coefficients in which the degrees need not all be the same. We provide sufficient conditions for which the system of equations $f_j (x_1, \ldots, x_n) = 0 \ (1…

Let $F(x,y)$ be an irreducible form of degree $r\geq 3$ and having $s+1$ non-zero coefficients. Let $h\geq 1$ be an integer and consider the Thue inequality $$|F(x,y)|\leq h.$$ Following the seminal work of Thue in 1909, several papers were…

Given an autonomous first order algebraic ordinary differential equation F(y,y')=0, we prove that every formal Puiseux series solution, expanded around any finite point or at infinity, is convergent. The proof is constructive and we provide…

Let $f(x,y)$ be a real polynomial of degree $d$ with isolated critical points, and let $i$ be the index of $grad f$ around a large circle containing the critical points. An elementary argument shows that $|i| \leq d-1$. In this paper we…

We give the first known bound for orders of differentiations in differential Nullstellensatz for both partial and ordinary algebraic differential equations. This problem was previously addressed by A. Seidenberg but no complete solution was…

Erd\"os and Obl\'ath proved that the equation $n!\pm m!=x^p$ has only finitely many integer solutions. More general, under the ABC-conjecture, Luca showed that $P(x)=An!+Bm!$ has finitely many integer solutions for polynomials of degree…

We study a natural complexity measure of Boolean functions known as the rational degree. Denoted $\textrm{rdeg}(f)$, it is the minimal degree of a rational function that is equal to $f$ on the Boolean hypercube. For total functions $f$, it…

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…