Counting Polynomial Roots in Isabelle/HOL: A formal Proof of the Budan-Fourier Theorem
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 root-counting 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, as an application, we apply the results 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.
Mon 14 JanDisplayed time zone: Belfast change
16:00 - 17:30 | Research Papers: Formalization of Mathematics and Computer AlgebraCPP at Sala XII Chair(s): Georges Gonthier Inria | ||
16:00 30mResearch paper | A Formal Proof of Hensel's Lemma over the p-adic Integers CPP Robert Y. Lewis Vrije Universiteit Amsterdam DOI | ||
16:30 30mResearch paper | Counting Polynomial Roots in Isabelle/HOL: A formal Proof of the Budan-Fourier Theorem CPP DOI | ||
17:00 30mResearch paper | Smooth Manifolds and Types to Sets for Linear Algebra in Isabelle/HOL CPP DOI |