On Solving Fewnomials Over Intervals in Fewnomial Time
J. Maurice Rojas, Yinyu Ye
TL;DR
This work advances real solving of univariate sparse polynomials by proving that when the polynomial is a trinomial ($m=3$), all real roots in a finite interval can be ε-approximate in time polylogarithmic in the degree, specifically $O(\log(D)\log(D\log(R/\varepsilon)))$, and the number of roots can be counted in $O(\log^2 D)$ arithmetic operations. The approach combines a refined α-theory framework with a hybrid bisection/Newton method and a Sturm-sequence-based counting strategy, implemented in the MNOMIALSOLVE algorithm with subroutines HYBRID and FASTERCOUNT. A key algebraic observation is that the Sturm sequence for trinomials is highly sparse, allowing efficient evaluation, and connections are drawn to A-discriminants to explain the tractability for small $m$. The paper also discusses extensions to general $m$, highlighting fundamental obstacles (Problems A and B) and situating the results in the context of Smale’s 17th problem and real solving in sparse polynomial systems.
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 eps all the roots of f in the interval [0,R] using just O(log(D)log(Dlog(R/eps))) arithmetic operations. In particular, we can count the number of roots in any bounded interval using just O(log^2 D) 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.