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We establish a novel upper bound for the real solutions of the equation specified in the title, employing a generalized derivation-division algorithm. As a consequence, we also derive a new set of Chebyshev functions adapted specifically…

We give two elementary proofs, at a level understandable by students with only pre-calculus knowledge of Algebra, of the well known fact that an irreducible irrational n-th root of a positive rational number cannot be solution of a…

We show that a linear differential equation whose coefficients are entire functions of completely regular growth may have an entire solution of finite order which is not of completely regular growth. This answers a question of Gol'dberg and…

An upper bound for the number of distinct eigenvalues of a perturbed matrix has been recently established by P. E. Farrell [1, Theorem 1.3]. The estimate is the central result in Farrell's work and can be applied to estimate the number of…

This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…

In this elementary note we prove that a polynomial with rational coefficients divides the derivative of some polynomial which splits in $\Q$ if and only if all of its irrational roots are real and simple. This provides an answer to a…

Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…

In this paper methods for simultaneous finding all roots of generalized polynomials are developed. These methods are related to the case when the roots are multiple. They possess cubic rate of convergence and they are as labour-consuming as…

We consider linear systems of recurrence equations whose coefficients are given in terms of indefinite nested sums and products covering, e.g., the harmonic numbers, hypergeometric products, $q$-hypergeometric products or their mixed…

We prove that there exist uniform $(+,\times,/)$-circuits of size $O(n^3)$ to compute the basis generating polynomial of regular matroids on $n$ elements. By tropicalization, this implies that there exist uniform $(\max,+,-)$-circuits and…

We study properties of coefficients of a linear form, originating from a multiple integral. As a corollary, we prove Vasilyev's conjecture, connected with the problem of irrationality of the Riemann zeta function at odd integers.

In this paper we derive an upper bound for the degree of the strict invariant algebraic curve of a polynomial system in the complex project plane under generic condition. The results are obtained through the algebraic multiplicities of the…

We show that, under reasonable assumptions, two negative roots can be eliminated from the characteristic equation of a polynomial-like iterative equation. This result gives a new case where we may lower the order of such an equation.

We extend the definition of $\mathcal{A}$-discriminant varieties, and Kapranov's parametrization of $\mathcal{A}$-discriminant varieties, to complex exponents. As an application, we study the special case where $\mathcal{A}$ is a fixed real…

We introduce a method for proving Sum-of-Squares (SoS)/ Lasserre hierarchy lower bounds when the initial problem formulation exhibits a high degree of symmetry. Our main technical theorem allows us to reduce the study of the positive…

We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate roots)…

We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address…

For each $n$, let $\text{RD}(n)$ denote the minimum $d$ for which there exists a formula for the general polynomial of degree $n$ in algebraic functions of at most $d$ variables. In 1945, Segre called for a better understanding of the large…

We obtain a small improvement of Gallagher's larger sieve and we extend it to higher dimensions. We also obtain two interesting upper bounds for the number of solutions to polynomial congruences.

A fundamental problem in the theory of linearized and projective polynomials over finite fields is to characterize the number of roots in the coefficient field directly from the coefficients. We prove results of this type, of a recursive…