相关论文: Univariate polynomials and the contractibility of …

F. Bergeron recently asked the intriguing question whether $\binom{b+c}{b}_q -\binom{a+d}{d}_q$ has nonnegative coefficients as a polynomial in $q$, whenever $a,b,c,d$ are positive integers, $a$ is the smallest, and $ad=bc$. We conjecture…

Let $f(q)=a_rq^r+\cdots+a_sq^s$, with $a_r\neq 0$ and $a_s\neq 0$, be a real polynomial. It is a palindromic polynomial of darga $n$ if $r+s=n$ and $a_{r+i}=a_{s-i}$ for all $i$. Polynomials of darga $n$ form a linear subspace…

Let $f(T)$ be a monic polynomial of degree $d$ with coefficients in a finite field $\mathbb{F}_q$. Extending earlier results in the literature, but now allowing $(q,2d)>1$, we give a criterion for $f$ to satisfy the following property: for…

We consider the set of monic irreducible polynomials $P$ over a finite field $\mathbb{F}_q$ such that the multiplicative order modulo $P$ of some a in $\mathbb{F}_q(T)$ is divisible by a fixed positive integer $d$. Call $R_q(a,d)$ this set.…

Connection of flat polynomials with some spectral questions in ergodic theory is discussed. A necessary condition for a sequence of polynomials of the type $\frac{1}{\sqrt{N}} \big(1 +\sum_{j=1}^{N-1} z^{n_j}\big)$ to be flat in almost…

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more…

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

The relationship between a polynomial's zeros and factors is well known. If a is a zero of f(x) then (x-a) is a factor of f(x). In this paper, we generalize this idea to polynomials of two variables and with real coefficients. We consider…

A univariate polynomial f over a field is decomposable if it is the composition f = g(h) of two polynomials g and h whose degree is at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and…

Let $K$ be an algebrically closed field and let $n\geq 1$. If $P\in K[X]=K[X_1,\ldots,X_n]$, $P\neq 0$, we denote by $I(P)$ the support of $P$, which is the finite subset of $\mathbb N^n$ such that $P=\sum_{i\in I(P)}a_iX^i$ with $a_i\in…

We investigate the computational complexity of deciding whether a given univariate integer polynomial p(x) has a factor q(x) satisfying specific additional constraints. When the only constraint imposed on q(x) is to have a degree smaller…

This paper presents some partial answers to the following question. QUESTION. If a normal space X is the union of an increasing sequence of open sets U(1), U(2), U(3) ... such that each U(n) contracts to a point in X, must X be…

We characterize compatible families of real-rooted polynomials, allowing both positive and negative leading coefficients. Our characterization naturally generalizes the same-sign characterization used by Chudnovsky and Seymour in their…

For every bivariate polynomial $p(z_1, z_2)$ of bidegree $(n_1, n_2)$, with $p(0,0)=1$, which has no zeros in the open unit bidisk, we construct a determinantal representation of the form $$p(z_1,z_2)=\det (I - K Z),$$ where $Z$ is an…

We characterize polynomials having the same set of nonzero cyclic resultants. Generically, for a polynomial $f$ of degree $d$, there are exactly $2^{d-1}$ distinct degree $d$ polynomials with the same set of cyclic resultants as $f$.…

For a real polynomial $p = \sum_{i=0}^{n} c_ix^i$ with no negative coefficients and $n\geq 6$, let $\beta (p) = \inf_{i=1}^{n-1} c_i^2/c_{i+1}c_{i-1}$ (so $\beta (p) \geq 1$ entails that $p$ is log concave). If $\beta(p) > 1.45...$, then…

For a polynomial $f(t) = 1+f_0t+\cdots +f_{d-1}t^d$ with positive integer coefficients Bell and Skandera ask if real rootedness of f(t) implies that there is a simplicial complex with f-vector $(1,f_0 \ldots,f_{d-1})$. In this paper we…

For a polynomial with palindromic coefficients, unimodality is equivalent to having a nonnegative $g$-vector. A sufficient condition for unimodality is having a nonnegative $\gamma$-vector, though one can have negative entries in the…

The generic monic polynomial of sixth degree features 6 a priori arbitrary coefficients. We show that if these 6 coefficients are appropriately defined in two different ways|in terms of 5 arbitrary parameters, then the 6 roots of the…

Let $K$ be a compact subset of a totally-real manifold $M$, where $M$ is either a $\mathcal{C}^2$-smooth graph in $\mathbb{C}^{2n}$ over $\mathbb{C}^n$, or $M=u^{-1}\{0\}$ for a $\mathcal{C}^2$-smooth submersion $u$ from $\mathbb{C}^n$ to…