Related papers: Generally rational polynomials in two variables

We study the rational approximation properties of special manifolds defined by a set of polynomials with rational coefficients. Mostly we will assume the case of all polynomials to depend on only one variable. In this case the manifold can…

An irreducible polynomial over $\Bbb F_q$ is said to be normal over $\Bbb F_q$ if its roots are linearly independent over $\Bbb F_q$. We show that there is a polynomial $h_n(X_1,\dots,X_n)\in\Bbb Z[X_1,\dots,X_n]$, independent of $q$, such…

Let f be a real or complex polynomial. We give an algorithm to compute the set of generalized critical values. The algorithm uses a finite dimensional space of rational arcs along which we can reach all generalized critical values of f.

In this paper we give several conditions implying the irreducibility of the algebraic curve P(x)-Q(y)=0, where P,Q are rational functions. We also apply the results obtained to the functional equations P(f)=Q(g) and P(f)=cP(g), where c\in…

An irreducible element of a commutative ring is absolutely irreducible if no power of it has more than one (essentially different) factorization into irreducibles. In the case of the ring $\text{Int}(D)=\{f\in K[x]\mid f(D)\subseteq D\}$,…

We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset $S$ of the algebra $\mathfrak g$ of left-invariant vector fields on a Lie group $\mathbb G$ and…

For each number field k, the Bombieri-Lang conjecture for k-rational points on surfaces of general type implies the existence of a polynomial f(x,y) in k[x,y] inducing an injection k x k --> k.

A formalism is given to count integer and rational solutions to polynomial equations with rational coefficients. These polynomials $P(x)$ are parameterized by three integers, labeling an elliptic curve. The counting of the rational…

Let $K$ be a field of characteristic $0$, and let $k \geq 2$ be an integer. We prove that every $K$-linear bijection $f \colon K[X] \to K[X]$ strongly preserving the set of $k$-free polynomials (or the set of polynomials with a $k$-fold…

Let $f$ and $g$ be Weierstrass polynomials with coefficients in the ring of formal power series over an algebraically closed field of characteristic zero. Assume that $f$ is irreducible and quasi-ordinary. We show that if degree of $g$ is…

Let $(P, Q)$ be a pair of Jacobian polynomials. We can show that $ <P, Q>+l+2g(P)-2= 0= <P, [P,Q]>$, where $<f, g>$ is the intersection number of $f, g\in \CC[x, y]$ in the affine plane, $l$ is the number of branch at point at infinity and…

In this paper we consider the curves $C_k^{(p,a)} : y^p-y=x^{p^k+1}+ax$ defined over $\mathbb F_p$ and give a positive answer to a conjecture about a divisibility condition on $L$-polynomials of the curves $C_k^{(p,a)}$. Our proof involves…

We characterize left and right amenable semigroups of polynomials of one complex variable with respect to the composition operation. We also prove a number of results about amenable semigroups of arbitrary rational functions. In particular,…

Let X be a geometrically rational (or more generally, separably rationally connected) variety over a finite field K. We prove that if K is large enough then X contains many rational curves defined over K. As a consequence we prove that…

We study properties of rational curves on complete intersections in positive characteristic. It has long been known that in characteristic 0, smooth Calabi-Yau and general type varieties are not uniruled. In positive characteristic,…

We study polynomials with no zeros on the unit ball in complex Euclidean space with a view toward characterizing when a rational function is bounded on the ball. We give a complete local description of such polynomials in two variables near…

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 $G$ be a finite 2-group and $K$ be a field satisfying that (i) $\fn{char}K\ne 2$, and (ii) $\sqrt{a}\in K$ for any $a\in K$. If $G$ acts on the rational function field $K(x,y,z)$ by monomial $K$-automorphisms, then the fixed field…

Some polynomials $P$ with rational coefficients give rise to well defined maps between cyclic groups, $\Z_q\longrightarrow\Z_r$, $x+q\Z\longmapsto P(x)+r\Z$. More generally, there are polynomials in several variables with tuples of rational…

This article deals with a quantitative aspect of Hilbert's seventeenth problem: producing a collection of real polynomials in two variables of degree 8 in one variable which are positive but are not a sum of three squares of rational…