Elsevier

Journal of Complexity

Volume 21, Issue 1, February 2005, Pages 87-110
Journal of Complexity

On solving univariate sparse polynomials in logarithmic time

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Abstract

Let f be a degree D univariate polynomial with real coefficients and exactly m monomial terms. We show that in the special case m=3 we can approximate within ε all the roots of f in the interval [0,R] using just O(log(D)log(DlogRε)) arithmetic operations. In particular, we can count the number of roots in any bounded interval using just O(log2D) arithmetic operations. Our speed-ups are significant and near-optimal: The asymptotically sharpest previous complexity upper bounds for both problems were super-linear in D, while our algorithm has complexity close to the respective complexity lower bounds. We also discuss conditions under which our algorithms can be extended to general m, and a connection to a real analogue of Smale's 17th Problem.

Keywords

Sparse polynomial
Lacunary polynomial
Computational real algebraic geometry
Fewnomial
Speed up
Trinomial
Newton's method
Smale's 17th Problem
Real root counting
m-nomial
Alpha theory
Discriminant
Logarithmic
Complexity
Approximate root

Cited by (0)

1

Partially funded by Hong Kong/France PROCORE Grant #9050140-730, a Texas A&M University Faculty of Science Grant, and NSF Grants DMS-0138446 and DMS-0211458.

2

Partially funded by NSF Grant DMS-9703490 and the City University of Hong Kong. The second author's work was done while visiting the Department of Mathematics at the City University of Hong Kong.