Use of algebraically independent numbers for zero recognition of polynomial terms

Daniel Richardson, Ahmed El-Sonbaty

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

A polynomial term is a tree with operators *, +, - on the interior nodes and natural numbers and variables on the frontier. We attempt to decide whether or not such a tree represents the zero polynomial by substituting algebraically independent real numbers for the variables and attempting to decide whether or not the resulting constant is zero. From this we get a probabilistic zero recognition test which is somewhat more expensive computationally than the usual method of choosing random integers in a large interval and evaluating, but which depends on the ability to choose a random point in the unit cube and to approximate a polynomial at that point. We also state a conjecture about algebraic independence measure which would give us a deterministic test with a uniformly chosen test point. The result is that if a polynomial term has s(T) nodes, then the bit complexity of deterministic zero recognition is bounded by O(s(T)M(s(T) length(T))), where length(T) measures the length of the term T, and M(n) is the bit complexity of multiplication of two n-digit natural numbers.
Original languageEnglish
Pages (from-to)631-637
Number of pages7
JournalJournal of Complexity
Volume19
Issue number5
DOIs
Publication statusPublished - 2003

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