Polynomiogram: An Integrated Framework for Root Visualization and Generative Art

This work presents the Polynomiogram framework, an integrated computational platform for exploring, visualizing, and generating art from polynomial root systems. The main innovation is a flexible sampling scheme in which two independent parameters are drawn from user defined domains and mapped to the polynomial coefficients through a generating function. This design allows the same mathematical foundation to support both scientific investigation and generative algorithmic art. The framework integrates two complementary numerical engines: NumPy companion matrix solver for fast, large scale computation and MPSolve for high precision, scientifically rigorous validation. This dual architecture enables efficient visualization for creative use and accurate computation for research and education. Numerical accuracy was verified using classical ensembles, including the Kac and Lucas polynomials. The method was applied to the cubic polynomial system to analyze its bifurcation structure, demonstrating its value as both a scientific tool for exploring root phenomena and an educational aid for visualizing fundamental concepts in algebra and dynamical systems. Beyond analysis, the Polynomiogram also demonstrated its potential as a tool for personalized generative art. Examples include the use of the platform to generate a natural form resembling a hibiscus flower and to create personalized artwork expressing gratitude toward advances in artificial intelligence and large language models through a tribute composition.