The aim of this paper is to study how efficiently we evaluate a system of multivariate polynomials and their partial derivatives in homotopy continuation methods. Our major tool is an extension of the Horner scheme, which is popular in evaluating a univariate polynomial, to a multivariate polynomial. But the extension is not unique, and there are many Horner factorizations of a given multivaria...

In my thesis, I use the planar algebra defined by Bigelow [4] to describe the multivariate Alexander polynomial. The advantage to using a planar algebra in this way is that the multivariate Alexander polynomial generalizes to a multivariate tangle invariant up to Reidemeister I. The planar algebra that I used in my thesis is the Motzkin planar algebra, which has connections to the representatio...

The aim of this paper is to study how efficiently we evaluate a system of multivariate polynomials and their partial derivatives in homotopy continuation methods. Our major tool is an extension of the Horner scheme, which is popular in evaluating a univariate polynomial, to a multivariate polynomial. But the extension is not unique, and there are many Horner factorizations of a given multivaria...

In this paper we develop a new approach for learning decision trees and multivariate polynomials via interpolation of multivariate polynomials. This new approach yields simple learning algorithms for multivariate polynomials and decision trees over nite elds under any constant bounded product distribution. The output hypothesis is a (single) multivariate polynomial that is an-approximation of t...