Evaluating Winding Numbers and Counting Complex Roots Through Cauchy Indices in Isabelle/HOL

Abstract

In complex analysis, the winding number measures the number of times a path (counter-clockwise) winds around a point, while the Cauchy index can approximate how the path winds. We formalise this approximation in the Isabelle theorem prover, and provide a tactic to evaluate winding numbers through Cauchy indices. By further combining this approximation with the argument principle, we are able to make use of remainder sequences to effectively count the number of complex roots of a polynomial within some domains, such as a rectangular box and a half-plane.

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

“…In the previous sections ( §3 and §4), we have demonstrated our enhancements for counting real roots in Isabelle/HOL. In this section, we will further apply those enhancements to improve existing complex-root-counting procedures [21]. In particular, we will first review the idea of counting complex roots through Cauchy indices in §5.1.…”

“…Index In this section we will briefly review the idea of counting complex roots through Cauchy indices, For a more detailed explanation, the reader can refer to our previous work [21].…”

“…Because it is declared as a code equation, it implements a verified procedure to compute Num C (P; {z | Im(z) > 0}). We can now type the following command in Isabelle/HOL: value proots_upper [:1+i, -2-i, 1:] to compute Num C ((1 + i) + (−2 − i)x + x 2 ; {z | Im(z) > 0}), which was not possible in our previous work [21] since the polynomial (1 + i) + (−2 − i)x + x 2 = (x − 1)(x − 1 − i) has a root on the border (i.e., the real axis).…”

“…Counting distinct real roots with Sturm's theorem has been widely implemented among major proof assistants including PVS [25], Coq [22], HOL Light [24] and Isabelle [11,15,18]. In contrast, our previous complex-root-counting procedure [21] seems to be the only one that counts complex roots, since counting complex roots usually requires a formal proof of the argument principle in complex analysis, which (to the best of our knowledge) is only available in Isabelle/HOL [20].…”

Efficient Construction of Voxel Models for Ore Bodies Using an Improved Winding Number Algorithm and CUDA Parallel Computing

“…While scholars have explored the discrimination of 3D non-convex polygon models with self-intersections, holes, or degeneracies [38,39], the Winding number remains a mathematical concept that is still being studied in complex analysis. In this field, it counts the number of complex roots of a polynomial [40]. Kodama et al [41,42] extended the Winding number algorithm into 3D space by summing solid angles to determine whether a point is inside or outside a nonconvex polyhedron.…”