Injectivity of polynomials over finite discrete dynamical systems
Antonio E. Porreca, Marius Rolland
TL;DR
The paper investigates injectivity of univariate polynomials over finite discrete-time dynamical systems (FDDS) by leveraging the unroll representation of transient dynamics. It proves that all univariate polynomials over unrolls are injective and introduces a polynomial-time, kroot-like algorithm to solve equations of the form $P(\mathcal{U}(X))=\mathcal{U}(B)$, leveraging a depth-cut deposition and a tree-order framework. A key result is that a polynomial is injective iff some non-constant coefficient is cancelable, with this condition shown to be both sufficient and, for FDDS, necessary; the constant-term case is handled by reduction. These findings enable efficient analysis and decomposition of FDDS behavior and open avenues for extending injectivity to multivariate polynomials and broader equation classes.
Abstract
The analysis of observable phenomena (for instance, in biology or physics) allows the detection of dynamical behaviors and, conversely, starting from a desired behavior allows the design of objects exhibiting that behavior in engineering. The decomposition of dynamics into simpler subsystems allows us to simplify this analysis (or design). Here we focus on an algebraic approach to decomposition, based on alternative and synchronous execution as the sum and product operations; this gives rise to polynomial equations (with a constant side). In this article we focus on univariate, injective polynomials, giving a characterization in terms of the form of their coefficients and a polynomial-time algorithm for solving the associated equations.