Related papers: On a stratification defined by real roots of polyn…

Theorem 1 is a formula expressing the mean number of real roots of a random multihomogeneous system of polynomial equations as a multiple of the mean absolute value of the determinant of a random matrix. Theorem 2 derives closed form…

For a smooth, projective family of homogeneous varieties defined over a number field, we show that if potential density holds for the rational points of the base, then it also holds for the total space. A conjecture of Campana and…

We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic…

We define the notion of complex stratification by quasifolds and show that such spaces occur as complex quotients by certain nonclosed subgroups of tori associated to convex polytopes. The spaces thus obtained provide a natural…

We obtain the list of automorphism groups for smooth plane sextic curves over an algebraically closed field K of characteristic p=0 or p>21. Moreover, we assign to each group a geometrically complete family over K describing its…

We show that the pattern $\{x,x+y,xy\}$ is partition regular over the space of formal integer polynomials of degree at least one with zero constant term, with primitive recursive bounds. This provides a new proof for the partition…

The well-known mathematical instrument for detection common roots for pairs of polynomials and multiple roots of polynomials are resultants and discriminants. For a pair of polynomials $f$ and $g$ their resultant $R(f,g)$ is a function of…

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…

We formulate a conjecture concerning spectral factorization of a class of trigonometric polynomials of two variables and prove it for special cases. Our method uses relations between the distribution of values of a polynomial of two…

The triangle of sorted binomial coefficients $\left\langle {n \atop k} \right\rangle = \binom{n}{\lfloor \frac{n - k}{2} \rfloor}$ for $0 \leq k \leq n$ has appeared several times in recent combinatorial works but has evaded dedicated…

We show a stability-type theorem for foliations on projective spaces which arise as pullbacks of foliations with a split tangent sheaf on weighted projective spaces. As a consequence, we will be able to construct many irreducible components…

For every polynomial f of degree n with no double roots, there is an associated family C(f) of harmonic algebraic curves, fibred over the circle, with at most n-1 singular fibres. We study the combinatorial topology of C(f) in the generic…

Let $G_n(z)=\xi_0+\xi_1z+...+\xi_n z^n$ be a random polynomial with i.i.d. coefficients (real or complex). We show that the arguments of the roots of $G_n(z)$ are uniformly distributed in $[0,2\pi]$ asymptotically as $n\to\infty$. We also…

In this note we consider algebraic exponential sums over the values of homogeneous nonsingular polynomials $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ in the quotient ring $\mathbb{Z}/p^2\mathbb{Z}$. We provide an estimate of…

A topological invariant of a polynomial map $p:X\to B$ from a complex surface containing a curve $C\subset X$ to a one-dimensional base is given by a rational second homology class in the compactification of the moduli space of genus $g$…

We combine the known methods for univariate polynomial root-finding and for computations in the Frobenius matrix algebra with our novel techniques to advance numerical solution of a univariate polynomial equation, and in particular…

We consider a new stratification of the space of configurations of $n$ marked points on the complex plane. Recall that this space can be differently interpreted as the space $^{\rm D}{\rm Pol}_{n}$ of degree $n>1$ complex, monic polynomials…

In this paper, we prove a number of results providing either necessary or sufficient conditions guaranteeing that the number of real roots of real polynomials of a given degree is either less or greater than a given number. We also provide…

In this short note we prove a couple of facts about polynomial count varieties, answering natural questions that they raise. A polynomial count $X$ variety is essentially one for which its number of points over finite fields is given by a…

We study the geometry of the stratification induced by an affine hyperplane arrangement H on the quotient of a complex affine space by the action of a discrete group preserving H. We give conditions ensuring normality or normality in…