Related papers: Real Zeros and Partitions without singleton blocks

We examine two binary operations on the set of algebraic polynomials, known as multiplicative and additive finite free convolutions, specifically in the context of hypergeometric polynomials. We show that the representation of a…

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…

We present six Theorems on the univariate real Polynomial, using which we develop a new algorithm for deciding the existence of atleast one real root for univariate integer Polynomials. Our algorithm outputs that no positive real root…

Let $N=2n^2-1$ or $N=n^2+n-1$, for any $n\ge 2$. Let $M=\frac{N-1}{2}$. We construct families of prime knots with Jones polynomials $(-1)^M\sum_{k=-M}^{M} (-1)^kt^k$. Such polynomials have Mahler measure equal to $1$. If $N$ is prime, these…

Partitions without sequences of consecutive integers as parts have been studied recently by many authors, including Andrews, Holroyd, Liggett, and Romik, among others. Their results include a description of combinatorial properties,…

The literature on quaternionic polynomials and, in particular, on methods for determining and classifying their zero-sets, is fast developing and reveals a growing interest on this subject. In contrast, polynomials defined over the algebra…

If the coefficients of polynomials are selected by some random process, the zeros of the resulting polynomials are in some sense random. In this paper the author rephrases the above in more precise language, and calculates the joint…

A novel method with two variations is proposed with which the number of positive and negative zeros of a polynomial with real coefficients and degree $n$ can be restricted with significantly better determinacy than that provided by the…

Given a natural number $n \geq 4$ we show that there exists infinitely many polynomials $f_{n}(x):= \prod_{i=1}^{n} (x^{2} - a_{i})$ such that (i) $f_{n}(x)$ has a root modulo every positive integer, (ii) $f_{n}(x)$ has no rational roots,…

We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients.…

We study the probability distribution of the number of zeros of multivariable polynomials with bounded degree over a finite field. We find the probability generating function for each set of bounded degree polynomials. In particular, in the…

The plane partition polynomial $Q_n(x)$ is the polynomial of degree $n$ whose coefficients count the number of plane partitions of $n$ indexed by their trace. Extending classical work of E.M. Wright, we develop the asymptotics of these…

We prove the classical result, which goes back at least to Fourier, that a polynomial with real coefficients has all zeros real and distinct if and only if the polynomial and also all of its nonconstant derivatives have only negative minima…

Consider a system $f_1(x)=0,\ldots,f_n(x)=0$ of $n$ random real polynomials in $n$ variables, where each $f_i$ has a prescribed set of terms described by a set $A\subseteq \mathbb{N}^n$ of cardinality $t$. Assuming that the coefficients of…

We introduce the concept of piecewise interlacing zeros for studying the relation of root distribution of two polynomials. The concept is pregnant with an idea of confirming the real-rootedness of polynomials in a sequence. Roughly…

We investigate non-unique factorization of polynomials in Z_{p^n}[x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, Z_{p^n}[x] is atomic. We reduce the question of factoring arbitrary…

Let $B$ be an infinite subset of $\mathbf{N}$. When we consider partitions of natural numbers into elements of $B$, a partition number without a restriction of the number of equal parts can be expressed by partition numbers with a…

A conjecture on the monotonicity of t-core partitions in an article of Stanton [Open positivity conjectures for integer partitions, Trends Math., 2:19-25, 1999] has been the catalyst for much recent research on t-core partitions. We…

Many statistics of roots of random polynomials have been studied in the literature, but not much is known on the concentration aspect. In this note we present a systematic study of this question, aiming towards nearly optimal bounds to some…

In this paper, we will derive the real roots of certain sets of matrices with real entries. We will also demonstrate that real orthogonal matrices can have real root or be involutory. Eventually, we will represent idempotent matrices in a…