In this paper a parametrized Newton-Kantorovich approach is applied to continuation of periodic orbits in arbitrary polynomial vector fields. This allows us rigorously validate numerically computed branches solutions. We derive the estimates full generality and present sample proofs obtained using an implementation Matlab. The presented applicable any field order di...

In this paper, we provide a quantitative estimate of unique continuation (doubling property) for higher-order parabolic partial differential equations with non-analytic Gevrey coefficients. Also, a new upper bound is given on the number of zeros for the solutions with a polynomial dependence on the coefficients.

0. Introduction and Preliminary 2 0.1. Existence and uniqueness theorem 2 0.2. Equation of a chain 7 1. Nonsingular matrices 9 1.1. A family of nonsingular matrices 9 1.2. Sufficient condition for Nonsingularity 12 1.3. Estimates 16 2. Local automorphism group of a real hypersurface 23 2.1. Polynomial Identities 23 2.2. Injectivity of a Linear Mapping 33 2.3. Beloshapka-Loboda Theorem 42 3. Com...

Complexity theoretic aspects of continuation methods for the solution of square or underdetermined systems of polynomial equations have been studied by various authors. In this paper we consider overdetermined systems where there are more equations than unknowns. We study Newton’s method for such a system.

Abstract. Since for q > 1, the q-Bernstein polynomials Bn,q are not positive linear operators on C[0, 1], the investigation of their convergence properties turns out to be much more difficult than that in the case 0 < q < 1. In this paper, new results on the approximation of continuous functions by the q-Bernstein polynomials in the case q > 1 are presented. It is shown that if f ∈ C[0, 1] and ...

We have developed a probabilistic method that uses numerical polynomial homotopy continuation to compute Galois groups of Schubert problems. Our implementation of the method produced answers to the questions considered inaccessible by other techniques. The problem of computing all solutions of a system of polynomial equations may be solved using numerical polynomial homotopy continuation [10]. ...

We obtain new necessary conditions for an n-dimensional semialgebraic subset of R to be a polynomial image of R. Moreover, we prove that a large family of planar bidimensional semialgebraic sets with piecewise linear boundary are images of polynomial or regular maps, and we estimate in both cases the dimension of their generic fibers.

By a numerical continuation method called a diagonal homotopy, one can compute the intersection of two irreducible positive dimensional solution sets of polynomial systems. This paper proposes to use this diagonal homotopy as the key step in a procedure to intersect general solution sets that are not necessarily irreducible or even equidimensional. Of particular interest is the special case whe...

The set of Nash equilibria of a finite game is the set of nonnegative solutions to a system of polynomial equations. In this survey article we describe how to construct certain special games and explain how to find all the complex roots of the corresponding polynomial systems, including all the Nash equilibria. We then explain how to find all the complex roots of the polynomial systems for arbi...

As a continuation of the e cient and accurate polynomial interpolation time-marching technique in one-dimensional and two-dimensional cases, this paper proposes a method to extend it to threedimensional problems. By stacking all two-dimensional (x, y) slice matrices along Z direction, threedimensional derivative matrices are constructed so that the polynomial interpolation time-marching can be ...