Cauchy index computation

Abstract

Computing the Cauchy index of a rational fractionQ/P betweena andb is important in most of the basic algorithms of real algebraic geometry: real root counting, exact sign determination, Routh-Hurwitz problem, signature of a Hankel matrix. One way to compute the Cauchy index ofQ/P is to compute some variant of the Euclidean remainder sequence ofP andQ and to compute the signs of the successive remainders evaluated ata andb. So, we want to compute efficiently the signs of the polynomials in the Euclidean remainder sequence evaluated at a given point. In the first part of the paper, we are only interested in counting arithmetic operations and comparisons. For input polynomials of degrees bounded byd we design an algorithm with complexity$O(\mathcal{M}(d)\log (d + 1))$ for computing the Cauchy index based on the strategy of a known efficient algorithm for computing a GCD; here$\mathcal{M}(d)$ denotes the cost of multiplying polynomials of degree at mostd. One has$\mathcal{M}(d) = O(d\log (d + 1))$ if the coefficient field allows Fourier transform, and$\mathcal{M}(d) = O(d\log (d + 1)\log \log (d + 2))$ otherwise. In the second part of the paper we are interested in the bit complexity when the coefficients ofP andQ are integers of size τ. A better way to compute the Cauchy index is then to evaluate the values of some variant of the subresultants rather than the values of the remainders. For polynomials of bit size τ we design an algorithm with bit complexity$O(\mathcal{M}(d,\sigma )\log (d + 1))$ with σ=O(dτ) where$\mathcal{M}(d,\sigma ) = O(d\sigma \cdot \log (d\sigma ) \cdot \log \log (d\sigma ))$ is Schönhage’s bound for multiplication of integer polynomials of degrees bounded byd and bit sizes bounded by σ in the multitape Turing machine model. So our bound isO(d ²τ·log(dτ)·log log(dτ)·log(d+1)). The same bound holds for computing the signature of a regular Hankel matrix. Our analysis shows a new and natural exact divisibility for subresultants.