A Generic Position Based Method for Real Root Isolation of Zero-Dimensional Polynomial Systems

We improve the local generic position method for isolating the real roots of a zero-dimensional bivariate polynomial system with two polynomials and extend the method to general zero-dimensional polynomial systems. The method mainly involves resultant computation and real root isolation of univariate polynomial equations. The roots of the system have a linear univariate representation. The complexity of the method is $\tilde{O}_B(N^{10})$ for the bivariate case, where $N=\max(d,\tau)$, $d$ resp., $\tau$ is an upper bound on the degree, resp., the maximal coefficient bitsize of the input polynomials. The algorithm is certified with probability 1 in the multivariate case. The implementation shows that the method is efficient, especially for bivariate polynomial systems.Comment: 24 pages, 5 figure

Univariate real root isolation in an extension field and applications

International audienceWe present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in $B_{\alpha} \in L[y]$, where L=\QQ(\alpha) is a simple algebraic extension of the rational numbers. We revisit two approaches for the problem. In the first approach, using resultant computations, we perform a reduction to a polynomial with integer coefficients and we deduce a bound of \sOB(N^{8}) for isolating the real roots of $B_{\alpha}$, where $N$ is an upper bound on all the quantities (degree and bitsize) of the input polynomials. The bound becomes \sOB(N^{7}) if we use Pan's algorithm for isolating the real roots. %% In the second approach we isolate the real roots working directly on the polynomial of the input. We compute improved separation bounds for the roots and we prove that they are optimal, under mild assumptions. For isolating the real roots we consider a modified Sturm algorithm, and a modified version of \func{descartes}' algorithm. For the former we prove a Boolean complexity bound of \sOB(N^{12}) and for the latter a bound of \sOB(N^{5}). %% We present aggregate separation bounds and complexity results for isolating the real roots of all polynomials $B_{\alpha_k}$, when $\alpha_k$ runs over all the real conjugates of $\alpha$. We show that we can isolate the real roots of all polynomials in \sOB(N^5). %% Finally, we implemented the algorithms in \func{C} as part of the core library of \mathematica and we illustrate their efficiency over various data sets

Theorem of three circles in Coq

The theorem of three circles in real algebraic geometry guarantees the termination and correctness of an algorithm of isolating real roots of a univariate polynomial. The main idea of its proof is to consider polynomials whose roots belong to a certain area of the complex plane delimited by straight lines. After applying a transformation involving inversion this area is mapped to an area delimited by circles. We provide a formalisation of this rather geometric proof in Ssreflect, an extension of the proof assistant Coq, providing versatile algebraic tools. They allow us to formalise the proof from an algebraic point of view.Comment: 27 pages, 5 figure

Real Root Isolation of Regular Chains

We present an algorithm RealRootIsolate for isolating the real roots of a system of multivariate polynomials given by a zerodimensional squarefree regular chain. The output of the algorithm is guaranteed in the sense that all real roots are obtained and are described by boxes of arbitrary precision. Real roots are encoded with a hybrid representation, combining a symbolic object, namely a regular chain, and a numerical approximation given by intervals. Our isolation algorithm is a generalization, for regular chains, of the algorithm proposed by Collins and Akritas. We have implemented RealRootIsolate as a command of the module SemiAlgebraicSetTools of the RegularChains library in Maple. Benchmarks are reported.

On the Complexity of Real Root Isolation

We introduce a new approach to isolate the real roots of a square-free polynomial $F=\sum_{i=0}^n A_i x^i$ with real coefficients. It is assumed that each coefficient of $F$ can be approximated to any specified error bound. The presented method is exact, complete and deterministic. Due to its similarities to the Descartes method, we also consider it practical and easy to implement. Compared to previous approaches, our new method achieves a significantly better bit complexity. It is further shown that the hardness of isolating the real roots of $F$ is exclusively determined by the geometry of the roots and not by the complexity or the size of the coefficients. For the special case where $F$ has integer coefficients of maximal bitsize $\tau$, our bound on the bit complexity writes as $\tilde{O}(n^3\tau^2)$ which improves the best bounds known for existing practical algorithms by a factor of $n=deg F$. The crucial idea underlying the new approach is to run an approximate version of the Descartes method, where, in each subdivision step, we only consider approximations of the intermediate results to a certain precision. We give an upper bound on the maximal precision that is needed for isolating the roots of $F$. For integer polynomials, this bound is by a factor $n$ lower than that of the precision needed when using exact arithmetic explaining the improved bound on the bit complexity

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