A numerical algorithm for zero counting, I: Complexity and accuracy

We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f. The algorithm performs O(log(nD(f))) iterations (grid refinements) where n is the number of polynomials (as well as the dimension of the ambient space), D is a bound on the polynomials' degree, and (f) is a condition number for the system. Each iteration uses an exponential number of operations. The algorithm uses finite-precision arithmetic and a major feature in our results is a bound for the precision required to ensure the returned output is correct which is polynomial in n and D and logarithmic in (f). The algorithm parallelizes well in the sense that each iteration can be computed in parallel time polynomial in n, log D and log((f)).

• Type: Journal
• Year: 2008
• Where: JC
• Authors: Felipe Cucker, Teresa Krick, Gregorio Malajovich, Mario Wschebor
• Keywords: Algorithm, Distinct Real Zeros, JC 2008, Polynomial