Algorithms for near solutions to polynomial equations

Algorithms for Computing Selected Solutions of Polynomial Equations

We present eecient and accurate algorithms to compute solutions of zero-dimensional multivariate polynomial equations in a given domain. The total number of solutions correspond to the Bezout bound for dense polynomial systems or the Bernstein bound for sparse systems. In most applications the actual number of solutions in the domain of interest is much lower than the Bezout or Bernstein bound....

Polynomial Solutions to Constant Coefficient Differential Equations

Let Dx, ... , Dr e C[d/dxx, ... , d/dxn) be constant coefficient differential operators with zero constant term. Let S = {fe C[xx,... , x„]\Dj(f) = 0 for all 1 < j < r) be the space of polynomial solutions to the system of simultaneous differential equations Dj(f) = 0. It is proved that S is a module over 3¡(V), the ring of differential operators on the affine scheme V with coordinate ring C[d/...

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...

Polynomial Solutions to Difference Equations Connected to Painleve’ Ii-vi

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...