Related papers: Generally rational polynomials in two variables

Let K be an algebraically closed field of characteristic zero and let f(x,y) be a nonzero polynomial of K[x,y]. We prove that if the generic element of the family $(f-\lambda)\_{\lambda}$ is a rational polynomial, and if the Jacobian J(f,g)…

Let K be an algebraically closed field of characteristic zero. Given a polynomial f(x,y) in K[x,y] with one place at infinity, we prove that either f is equivalent to a coordinate, or the family (f+c) has at most two rational elements. When…

Let k be an algebraically closed field of characteristic zero. An element F from k(x_1,...,x_n) is called a closed rational function if the subfield k(F) is algebraically closed in the field k(x_1,...,x_n). We prove that a rational function…

We find all polynomials f,g,h over a field K such that g and h are linear and f(g(x))=h(f(x)). We also solve the same problem for rational functions f,g,h, in case the field K is algebraically closed.

The main purpose of this paper is solve polynomial equations that are satisfied by (generalized) polynomials. More exactly, we deal with the following problem: let $\mathbb{F}$ be a field with $\mathrm{char}(\mathbb{F})=0$ and $P\in…

We are concerned with the behavior of the polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ with finite fibres and satisfying the condition that all of the curves $aP+bQ=0$, $(a:b)\in \mathbb{P}^1$, are irreducible rational curves. The obtained…

We classify the pairs of polynomials $f,g$ over a field $K$, such that $f(X)-g(Y)$ has a factor of total degree at most 2. This was done by Y. Bilu for characteristic 0 fields $K$. As his method does not work in positive characteristic, we…

A non-zero constant Jacobian polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ is invertible if $P$ and $Q$ are rational polynomials.

It has been proved several times in the literature that a polynomial map from $C^2$ to $C$ with irreducible rational fibers cannot be a component of a counterexample to the Jacobian Conjecture. This note points out that this result is…

Let $F(x,y)$ be a polynomial over the rationals. We show that if $F$ is not an expander (over the rationals) then it has a special multiplicative or additive form. For example if $F$ is a homogeneous non-expander polynomial then…

Let $k$ be an algebraically closed field of characteristic zero and $P(x,y)\in k[x,y]$ be a polynomial which depends on all its variables. $P$ has an algebraic constraint if the set $\{(P(a,b),(P(a',b'),P(a',b),P(a,b')\,|\,a,a',b,b'\in k\}$…

We describe algebraic curves $ X : F(x, y) = 0 $ defined over $\overline{\mathbb{Q}}$ that satisfy the following property: there exist a number field $k$ and an infinite set $S \subset k$ such that, for every $y \in S$, the roots of the…

Let G be a connected linear algebraic group over an algebraically closed field k, and let H be a connected closed subgroup of G. We prove that the homogeneous variety G/H is a rational variety over k whenever H is solvable, or when dim(G/H)…

The functional equation f(p(z))=g(q(z)) is studied, where p,q are polynomials and f,g are trancendental meromorphic functions in C. We find all the pairs p,q for which there exist nonconstant f,g satisfying our equation and there exist no…

For every algebraically closed field $\boldsymbol k$ of characteristic different from $2$, we prove the following: (1) Generic finite dimensional (not necessarily associative) $\boldsymbol k$-algebras of a fixed dimension, considered up to…

Let $A$ be a retract of the polynomial ring in three variables over a field $k$. It is known that if ${\rm char}\: (k) = 0$ or ${\rm tr.deg}\:_k A \not= 2$ then $A$ is a polynomial ring. In this paper, we give some sufficient conditions for…

Let F be a homogeneous real polynomial of even degree in any number of variables. We consider the problem of giving explicit conditions on the coefficients so that F is positive definite or positive semi-definite. In this note we produce a…

Consider the polynomial $f(x,y)=xy^k+C$ for $k\geq 2$ and any nonzero integer constant $C$. We derive an asymptotic formula for the $k$-free values of $f(x,y)$ when $x, y\leq H$. We also prove a similar result for the $k$-free values of…

A $k$-nucleus of a normal rational curve in PG$(n,F)$ is the intersection over all $k$-dimensional osculating subspaces of the curve ($k\in\{-1,0,...,n-1\}$). It is well known that for characteristic zero all nuclei are empty. In case of…

There exist homogeneous polynomials $f$ with $\mathbb Q$-coefficients that are sums of squares over $\mathbb R$ but not over $\mathbb Q$. The only systematic construction of such polynomials that is known so far uses as its key ingredient…