TY - JOUR

T1 - Use of algebraically independent numbers for zero recognition of polynomial terms

AU - Richardson, Daniel

AU - El-Sonbaty, Ahmed

N1 - ID number: ISI:000185363300001

PY - 2003

Y1 - 2003

N2 - 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.

AB - 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.

UR - http://dx.doi.org/10.1016/s0885-064x(03)00047-5

U2 - 10.1016/s0885-064x(03)00047-5

DO - 10.1016/s0885-064x(03)00047-5

M3 - Article

VL - 19

SP - 631

EP - 637

JO - Journal of Complexity

JF - Journal of Complexity

SN - 0885-064X

IS - 5

ER -