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

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

Bounded-degree factors of lacunary multivariate polynomials

In this paper, we present a new method for computing bounded-degree factors of lacunary multivariate polynomials. In particular for polynomials over number fields, we give a new algorithm that takes as input a multivariate polynomial f in lacunary representation and a degree bound d and computes the irreducible factors of degree at most d of f in time polynomial in the lacunary size of f and in d. Our algorithm, which is valid for any field of zero characteristic, is based on a new gap theorem that enables reducing the problem to several instances of (a) the univariate case and (b) low-degree multivariate factorization. The reduction algorithms we propose are elementary in that they only manipulate the exponent vectors of the input polynomial. The proof of correctness and the complexity bounds rely on the Newton polytope of the polynomial, where the underlying valued field consists of Puiseux series in a single variable.Comment: 31 pages; Long version of arXiv:1401.4720 with simplified proof

How many zeros of a random sparse polynomial are real?

We investigate the number of real zeros of a univariate $k$-sparse polynomial $f$ over the reals, when the coefficients of $f$ come from independent standard normal distributions. Recently B\"urgisser, Erg\"ur and Tonelli-Cueto showed that the expected number of real zeros of $f$ in such cases is bounded by $O(\sqrt{k} \log k)$. In this work, we improve the bound to $O(\sqrt{k})$ and also show that this bound is tight by constructing a family of sparse support whose expected number of real zeros is lower bounded by $\Omega(\sqrt{k})$. Our main technique is an alternative formulation of the Kac integral by Edelman-Kostlan which allows us to bound the expected number of zeros of $f$ in terms of the expected number of zeros of polynomials of lower sparsity. Using our technique, we also recover the $O(\log n)$ bound on the expected number of real zeros of a dense polynomial of degree $n$ with coefficients coming from independent standard normal distributions

Computing Real Roots of Real Polynomials

Computing the roots of a univariate polynomial is a fundamental and long-studied problem of computational algebra with applications in mathematics, engineering, computer science, and the natural sciences. For isolating as well as for approximating all complex roots, the best algorithm known is based on an almost optimal method for approximate polynomial factorization, introduced by Pan in 2002. Pan's factorization algorithm goes back to the splitting circle method from Schoenhage in 1982. The main drawbacks of Pan's method are that it is quite involved and that all roots have to be computed at the same time. For the important special case, where only the real roots have to be computed, much simpler methods are used in practice; however, they considerably lag behind Pan's method with respect to complexity. In this paper, we resolve this discrepancy by introducing a hybrid of the Descartes method and Newton iteration, denoted ANEWDSC, which is simpler than Pan's method, but achieves a run-time comparable to it. Our algorithm computes isolating intervals for the real roots of any real square-free polynomial, given by an oracle that provides arbitrary good approximations of the polynomial's coefficients. ANEWDSC can also be used to only isolate the roots in a given interval and to refine the isolating intervals to an arbitrary small size; it achieves near optimal complexity for the latter task.Comment: to appear in the Journal of Symbolic Computatio

Computing Real Roots of Real Polynomials -- An Efficient Method Based on Descartes' Rule of Signs and Newton Iteration

Computing the real roots of a polynomial is a fundamental problem of computational algebra. We describe a variant of the Descartes method that isolates the real roots of any real square-free polynomial given through coefficient oracles. A coefficient oracle provides arbitrarily good approximations of the coefficients. The bit complexity of the algorithm matches the complexity of the best algorithm known, and the algorithm is simpler than this algorithm. The algorithm derives its speed from the combination of Descartes method with Newton iteration. Our algorithm can also be used to further refine the isolating intervals to an arbitrary small size. The complexity of root refinement is nearly optimal

On Flows, Paths, Roots, and Zeros

This thesis has two parts; in the first of which we give new results for various network flow problems. (1) We present a novel dual ascent algorithm for min-cost flow and show that an implementation of it is very efficient on certain instance classes. (2) We approach the problem of numerical stability of interior point network flow algorithms by giving a path following method that works with integer arithmetic solely and is thus guaranteed to be free of any nu-merical instabilities. (3) We present a gradient descent approach for the undirected transship-ment problem and its special case, the single source shortest path problem (SSSP). For distrib-uted computation models this yields the first SSSP-algorithm with near-optimal number of communication rounds. The second part deals with fundamental topics from algebraic computation. (1) We give an algorithm for computing the complex roots of a complex polynomial. While achieving a com-parable bit complexity as previous best results, our algorithm is simple and promising to be of practical impact. It uses a test for counting the roots of a polynomial in a region that is based on Pellet's theorem. (2) We extend this test to polynomial systems, i.e., we develop an algorithm that can certify the existence of a k-fold zero of a zero-dimensional polynomial system within a given region. For bivariate systems, we show experimentally that this approach yields signifi-cant improvements when used as inclusion predicate in an elimination method.Im ersten Teil dieser Dissertation präsentieren wir neue Resultate für verschiedene Netzwerkflussprobleme. (1)Wir geben eine neue Duale-Aufstiegsmethode für das Min-Cost-Flow- Problem an und zeigen, dass eine Implementierung dieser Methode sehr effizient auf gewissen Instanzklassen ist. (2)Wir behandeln numerische Stabilität von Innere-Punkte-Methoden fürNetwerkflüsse, indem wir eine solche Methode angeben die mit ganzzahliger Arithmetik arbeitet und daher garantiert frei von numerischen Instabilitäten ist. (3) Wir präsentieren ein Gradienten-Abstiegsverfahren für das ungerichtete Transshipment-Problem, und seinen Spezialfall, das Single-Source-Shortest-Problem (SSSP), die für SSSP in verteilten Rechenmodellen die erste mit nahe-optimaler Anzahl von Kommunikationsrunden ist. Der zweite Teil handelt von fundamentalen Problemen der Computeralgebra. (1) Wir geben einen Algorithmus zum Berechnen der komplexen Nullstellen eines komplexen Polynoms an, der eine vergleichbare Bitkomplexität zu vorherigen besten Resultaten hat, aber vergleichsweise einfach und daher vielversprechend für die Praxis ist. (2)Wir erweitern den darin verwendeten Pellet-Test zum Zählen der Nullstellen eines Polynoms auf Polynomsysteme, sodass wir die Existenz einer k-fachen Nullstelle eines Systems in einer gegebenen Region zertifizieren können. Für bivariate Systeme zeigen wir experimentell, dass eine Integration dieses Ansatzes in eine Eliminationsmethode zu einer signifikanten Verbesserung führt