Counting polynomial roots in Isabelle/HOL: a formal proof of the Budan-Fourier theorem

Abstract

Many problems in computer algebra and numerical analysis can be reduced to counting or approximating the real roots of a polynomial within an interval. Existing verified rootcounting procedures in major proof assistants are mainly based on the classical Sturm theorem, which only counts distinct roots.In this paper, we have strengthened the root-counting ability in Isabelle/HOL by first formally proving the Budan-Fourier theorem. Subsequently, based on Descartes' rule of signs and Taylor shift, we have provided a verified procedure to efficiently over-approximate the number of real roots within an interval, counting multiplicity. For counting multiple roots exactly, we have extended our previous formalisation of Sturm's theorem. Finally, we combine verified components in the developments above to improve our previous certified complex-root-counting procedures based on Cauchy indices. We believe those verified routines will be crucial for certifying programs and building tactics.

A Verified Decision Procedure for Univariate Real Arithmetic with the BKR Algorithm

“…Conversely, it performs poorly on problems with many factors and higher degrees, e.g., ex3, ex6, and ex7. Further, as noted in experiments by Li and Paulson [16], the Sturm-Tarski theorem in Isabelle/HOL currently uses a straightforward method for computing remainder sequences which can also lead to significant (exponential) blowup in the bitsize of rational coefficients of the involved polynomials. This is especially apparent for ex6 and ex7, which have large polynomial degrees and high coefficient complexity; these time out without completing even a single Tarski query.…”