Related papers: Key polynomials for simple extensions of valued fi…

Using Okounkov's $q$-integral representation of Macdonald polynomials we construct an infinite sequence $\Omega_1,\Omega_2,\Omega_3,\dots$ of countable sets linked by transition probabilities from $\Omega_N$ to $\Omega_{N-1}$ for each…

Let $\mathbb{F}_q$ be a finite field with $q$ elements, $f \in \mathbb{F}_q[x_1, \dots, x_n]$ a polynomial in $n$ variables and let us denote by $N(f)$ the number of roots of $f$ in $\mathbb{F}_q^n$. %Many authors, such as Wei Cao and Kung…

Given a global field $K$ and a positive integer $n$, we present a diophantine criterion for a polynomial in one variable of degree $n$ over $K$ not to have any root in $K$. This strengthens the known result that the set of non-$n$-th-powers…

We show that the action of classical operators associated to the Macdonald polynomials on the basis of Schur functions, S_{\lambda}[X(t-1)/(q-1)], can be reduced to addition in \lambda-rings. This provides explicit formulas for the…

We consider the theory of algebraically closed fields of characteristic zero with multivalued operations $x\mapsto x^r$ (raising to powers). It is in fact the theory of equations in exponential sums. In an earlier paper we have described…

We provide an irreducibility test in the ring K[[x]][y] whose complexity is quasi-linear with respect to the valuation of the discriminant, assuming the input polynomial F square-free and K a perfect field of characteristic zero or greater…

For an integer $r$, a prime power $q$, and a polynomial $f$ over a finite field ${\mathbb F}_{q^r}$ of $q^r$ elements, we obtain an upper bound on the frequency of elements in an orbit generated by iterations of $f$ which fall in a proper…

Let K be a number field, and let lambda(x,t)\in K[x, t] be irreducible over K(t). Using algebraic geometry and group theory, we study the set of alpha\in K for which the specialized polynomial lambda(x,alpha) is K-reducible. We apply this…

We introduce orthogonal polynomials $M_j^{\mu,\ell}(x)$ as eigenfunctions of a certain self-adjoint fourth order differential operator depending on two parameters $\mu\in\mathbb{C}$ and $\ell\in\mathbb{N}_0$. These polynomials arise as…

Consider a system of polynomials in many variables over the ring of integers of a number field $K$. We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any…

Let $\mathbb{K}$ be the algebraic closure of a finite field $\mathbb{F}_q$ of odd characteristic $p$. For a positive integer $m$ prime to $p$, let $F=\mathbb{K}(x,y)$ be the transcendency degree $1$ function field defined by…

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,…

Let $K$ be a field of characteristic $p>0$, $A=K[[Y]]$ be a power series ring in one variable and $Q(A)$ be the field of fraction of $A$. Suppose that $R=A[X_1,\ldots,X_n]$ is a standard $\mathbb{N}^n$-graded polynomial ring over $A$, i.e.,…

Let K be a field and t>=0. Denote by Bm(t,K) the maximum number of non-zero roots in K, counted with multiplicities, of a non-zero polynomial in K[x] with at most t+1 monomial terms. We prove, using an unified approach based on Vandermonde…

It is well known that ordered exponential fields with a compatible non-trivial valuation cannot be spherically complete, but there are some that are ``complete enough''. This paper gives analogues of Kaplansky's theorem on maximally valued…

We establish a fundamental theorem of orders (FTO) which allows us to express all orders uniquely as an intersection of `irreducible orders' along which the index and the conductor distributes multiplicatively. We define a subclass of…

Let k be an algebraically closed field. Given an extension A : B of finite-dimensional k- algebras, we establish criteria ensuring that the representation-theoretic notion of polynomial growth is preserved under ascent and descent. These…

Higher-order Fourier analysis, developed over prime fields, has been recently used in different areas of computer science, including list decoding, algorithmic decomposition and testing. We extend the tools of higher-order Fourier analysis…

We study the number of prime polynomials of degree $n$ over $\mathbb{F}_q$ in which the $i^{th}$ coefficient is either preassigned to be $a_i \in \mathbb{F}_q$ or outside a small set $S_i \subset \mathbb{F}_q$. This serves as a function…

A simple recursive expansion algorithm for the integrals of tree level superstring five point amplitudes in a flat background is given which reduces the expansion to simple symbol(ic) manipulations. This approach can be used for instance to…