Counting positive solutions for polynomial systems with real coefficients

Counting Solutions to Polynomial Systems via Reductions

This paper provides both positive and negative results for counting solutions to systems of polynomial equations over a finite field. The general idea is to try to reduce the problem to counting solutions to a single polynomial, where the task is easier. In both cases, simple methods are utilized that we expect will have wider applicability (far beyond algebra). First, we give an efficient dete...

Counting Stable Solutions of Sparse Polynomial Systems

Small solutions to systems of polynomial equations with integer coefficients

The paper discusses a series of conjectures due to A. Tyszka aiming to describe boxes in which there exists at least one solution to a system of polynomial equations with integer coefficients. A proof of the bound valid in the linear case is given. 1 Two basic questions When facing systems of equations whose solutions are hard to determine, one is satisfied to determine (or at least estimate) t...

Existence of Positive Solutions for Multi-term Non-autonomous Fractional Differential Equations with Polynomial Coefficients

In the present paper we discuss the existence of positive solutions in the case of multi-term non-autonomous fractional differential equations with polynomial coefficients; the constant coefficient case has been studied in [2]. We consider the equation “ Dn − n−1 X j=1 pj(x)D αn−j ” y = f(x, y). We state various conditions on f and pj ’s under which this equation has: positive solutions, a uniq...

Existence of Positive Solutions for Multiterm Fractional Differential Equations of Finite Delay with Polynomial Coefficients

and Applied Analysis 3 Definition 2.2 see 15 . A cone K is called normal, if there exists a positive constant r such that f, g ∈ K and θ ≺ f ≺ g implies ‖f‖ ≤ r‖g‖, where θ denotes the zero element of K. Definition 2.3 see 16, 17 . Let f : a, b → R, and f ∈ L1 a, b . The left-sided RiemannLiouville fractional integral of f of order α is defined as I af x 1 Γ α ∫x a x − t α−1f t dt, α > 0, x ∈ a...