The number of polynomial solutions of polynomial Riccati equations

extensions of some polynomial inequalities to the polar derivative

توسیع تعدادی از نامساوی های چند جمله ای در مشتق قطبی

Bounds on the number of real solutions to polynomial equations

We use Gale duality for polynomial complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound e + 3 4 2( k 2)nk for the number of non-zero real solutions to a system of n polynomials in n variables having n+k+1 monomials whose exponent vectors generate a subgroup of Z of odd index. This bound exceeds the bound for positive solutions only by the ...

Improving the Accuracy of the Solutions of Riccati Equations

Polynomial solutions of differential equations

A new approach for investigating polynomial solutions of differential equations is proposed. It is based on elementary linear algebra. Any differential operator of the form L(y) = k=N ∑ k=0 ak(x)y, where ak is a polynomial of degree ≤ k, over an infinite ground field F has all eigenvalues in F in the space of polynomials of degree at most n, for all n. If these eigenvalues are distinct, then th...

Univariate Polynomial Solutions of Nonlinear Polynomial Difference Equations

We study real-polynomial solutions P (x) of difference equations of the formG(P (x−τ1), . . . , P (x− τs)) +G0(x)=0, where τi are real numbers, G(x1, . . . , xs) is a real polynomial of a total degree D ≥ 2, and G0(x) is a polynomial in x. We consider the following problem: given τi, G and G0, find an upper bound on the degree d of a real-polynomial solution P (x), if exists. We reduce this pro...