A Root Isolation Algorithm for Sparse Univariate Polynomials

8 double pages.International audienceWe consider a univariate polynomial f with real coefficients having a high degree $N$ but a rather small number $d+1$ of monomials, with $d\ll N$. Such a sparse polynomial has a number of real root smaller or equal to $d$. Our target is to find for each real root of $f$ an interval isolating this root from the others. The usual subdivision methods, relying either on Sturm sequences or Moebius transform followed by Descartes's rule of sign, destruct the sparse structure. Our approach relies on the generalized Budan-Fourier theorem of Coste, Lajous, Lombardi, Roy and the techniques developed in some previous works of Galligo. To such a $f$ is associated a set of $d + 1$ $\mathbb{F}$-derivatives. The Budan-Fourier function $V_f(x)$ counts the sign changes in the sequence of $\mathbb{F}$-derivatives of the $f$ evaluated at $x$. The values at which this function jumps are called the $\mathbb{F}$-virtual roots of $f$, these include the real roots of $f$. We also consider the augmented $\mathbb{F}$-virtual roots of $f$ and introduce a genericity property which eases our study. We present a real root isolation method and an algorithm which has been implemented in Maple. We rely on an improved generalized Budan-Fourier count applied to both the input polynomial and its reciprocal, together with Newton like approximation steps

Efficiently Computing Real Roots of Sparse Polynomials

We propose an efficient algorithm to compute the real roots of a sparse polynomial $f\in\mathbb{R}[x]$ having $k$ non-zero real-valued coefficients. It is assumed that arbitrarily good approximations of the non-zero coefficients are given by means of a coefficient oracle. For a given positive integer $L$, our algorithm returns disjoint disks $\Delta_{1},\ldots,\Delta_{s}\subset\mathbb{C}$, with $s<2k$, centered at the real axis and of radius less than $2^{-L}$ together with positive integers $\mu_{1},\ldots,\mu_{s}$ such that each disk $\Delta_{i}$ contains exactly $\mu_{i}$ roots of $f$ counted with multiplicity. In addition, it is ensured that each real root of $f$ is contained in one of the disks. If $f$ has only simple real roots, our algorithm can also be used to isolate all real roots. The bit complexity of our algorithm is polynomial in $k$ and $\log n$, and near-linear in $L$ and $\tau$, where $2^{-\tau}$ and $2^{\tau}$ constitute lower and upper bounds on the absolute values of the non-zero coefficients of $f$, and $n$ is the degree of $f$. For root isolation, the bit complexity is polynomial in $k$ and $\log n$, and near-linear in $\tau$ and $\log\sigma^{-1}$, where $\sigma$ denotes the separation of the real roots

A Near-Optimal Algorithm for Computing Real Roots of Sparse Polynomials

Let $p\in\mathbb{Z}[x]$ be an arbitrary polynomial of degree $n$ with $k$ non-zero integer coefficients of absolute value less than $2^\tau$. In this paper, we answer the open question whether the real roots of $p$ can be computed with a number of arithmetic operations over the rational numbers that is polynomial in the input size of the sparse representation of $p$. More precisely, we give a deterministic, complete, and certified algorithm that determines isolating intervals for all real roots of $p$ with $O(k^3\cdot\log(n\tau)\cdot \log n)$ many exact arithmetic operations over the rational numbers. When using approximate but certified arithmetic, the bit complexity of our algorithm is bounded by $\tilde{O}(k^4\cdot n\tau)$, where $\tilde{O}(\cdot)$ means that we ignore logarithmic. Hence, for sufficiently sparse polynomials (i.e. $k=O(\log^c (n\tau))$ for a positive constant $c$), the bit complexity is $\tilde{O}(n\tau)$. We also prove that the latter bound is optimal up to logarithmic factors