polynomialscan

polynomialscan is a software toolkit designed to analyze polynomials by scanning coefficients, parameter spaces, and variable domains to extract mathematical properties and approximate solutions. It combines symbolic manipulation with numeric methods to provide a unified workflow for root finding, factorization, and qualitative analysis of polynomials.

Core capabilities include real and complex root finding, real-root isolation, polynomial factorization over integers and rationals, sign analysis, monotonicity intervals derived from derivatives, and multiplicity detection. It supports parameterized polynomials, enabling users to track how roots and factors move as coefficients vary, by performing coefficient scans and continuity checks.

Implementation uses a hybrid approach: symbolic algorithms such as the Euclidean algorithm and factorization over Z for exact results; numerical methods for approximate roots (Durand–Kerner, Aberth, or companion-matrix eigenvalue methods); and Sturm sequences or Descartes' rule of signs for real-root isolation. The toolkit can operate on polynomials of high degree and with arbitrary precision arithmetic.

polynomialscan provides a programming API (for example in Python) and optional backends in C for performance; input formats include standard coefficient lists, symbolic expressions, and parameterized representations. Output can be returned as structured objects or serialized to JSON, enabling integration with other math software and notebooks.

Applications include computer algebra education, control theory for characteristic polynomials, numerical analysis research, and algebraic geometry explorations of root loci.

Limitations include numerical sensitivity for high-degree polynomials, potential combinatorial complexity in exact factorization, and reliance on well-conditioned inputs.

See also: polynomial, root-finding algorithms, Sturm's theorem, factorization over integers, companion matrix.