The vanishing ideal of a finite set of points with multiplicity structures

Given a finite set of arbitrarily distributed points in affine space with arbitrary multiplicity structures, we present an algorithm to compute the reduced Groebner basis of the vanishing ideal under the lexicographic ordering. Our method discloses the essential geometric connection between the relative position of the points with multiplicity structures and the quotient basis of the vanishing ideal, so we will explicitly know the set of leading terms of elements of I. We split the problem into several smaller ones which can be solved by induction over variables and then use our new algorithm for intersection of ideals to compute the result of the original problem. The new algorithm for intersection of ideals is mainly based on the Extended Euclidean Algorithm.Comment: 12 pages,12 figures,ASCM 201

Zeros of regular functions of quaternionic and octonionic variable: a division lemma and the camshaft effect

We study in detail the zero set of a regular function of a quaternionic or octonionic variable. By means of a division lemma for convergent power series, we find the exact relation existing between the zeros of two octonionic regular functions and those of their product. In the case of octonionic polynomials, we get a strong form of the fundamental theorem of algebra. We prove that the sum of the multiplicities of zeros equals the degree of the polynomial and obtain a factorization in linear polynomials.Comment: Proof of Lemma 7 rewritten (thanks to an anonymous reviewer

From Algebraic Sets to Monomial Linear Bases By Means of Combinatorial Algorithms.

Introduction 1.1 Let N be the monoid of non-negative integers. Denote by i := (i 1 ; : : : ; i n ) an arbitrary element in the power N n . The usual order on N, as well as the partial order it induces on N n , will be denoted by . Define an n-dimensional Ferrers diagram to be any finite ideal of the poset N n , i.e. any non-empty finite subset F ` N n such that j ! i 2 F =) j 2 F . An element i = (i 1 ; : : : ; i n ) 62 F is said to be a co-minimal element for the Ferrers diagram F if it is a minimal element of the complementary filter N n

Polynomial maps which are roots of power series

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