[Paper Review] An Empirical Method for Solving (rigorously!) Algebraic Functional Equations Of the Form F(P(x,t), P(x,1),x,t)=0

Ira M. Gessel, Doron Zeilberger|arXiv (Cornell University)|Dec 29, 2014

Functional Equations Stability Results3 references3 citations

TL;DR

This paper presents a rigorous, computer-assisted method for solving algebraic-functional equations of the form $ F(P(x,t), P(x,1), x, t) ≈ 0 $, where $ P(x,t) $ is a generating function and $ P(x,1) $ is its evaluation at $ t=1 $. By empirically guessing a bivariate polynomial $ I(Q(x), x) = 0 $ that annihilates $ P(x,1) $, then eliminating $ Q(x) $ via the resultant, the method derives an algebraic equation for $ P(x,t) $, enabling fully automatic and rigorous verification using symbolic computation tools like Maple.

ABSTRACT

We present an empirical-yet-rigorous approach for solving a wide class of functional equations, thereby automating many results that previously required considerable human ingenuity and human labor.

Motivation & Objective

• To develop a fully automatic, rigorous method for solving algebraic-functional equations arising in combinatorics, particularly those with catalytic variables.
• To overcome the limitations of the kernel method, which is labor-intensive and requires significant human ingenuity.
• To enable the automatic derivation of algebraic equations for bivariate generating functions $ P(x,t) $ from known univariate evaluations $ P(x,1) $.
• To implement the method in a publicly available Maple package, FunEq, for broad reproducibility and application.
• To demonstrate that empirical guessing, when combined with symbolic elimination, yields fully rigorous results without ad hoc human intervention.

Proposed method

• The method begins by empirically guessing a bivariate polynomial $ I(Q(x), x) $ such that $ I(P(x,1), x) \equiv 0 $, where $ P(x,1) $ is the univariate generating function.
• Using the functional equation $ F(P(x,t), P(x,1), x, t) \equiv 0 $, the method eliminates $ P(x,1) $ via the resultant operation with respect to $ Q(x) $, yielding an algebraic equation $ G(P(x,t), x, t) \equiv 0 $.
• The resultant computation ensures that the derived equation for $ P(x,t) $ is algebraically consistent with both the original functional equation and the known univariate relation.
• The method leverages the uniqueness of formal power series solutions to guarantee that if the derived equation holds and matches initial terms, it is correct by construction.
• The entire process is automated using symbolic computation, particularly Maple’s `resultant` command, and is fully rigorous once the initial guess is made.
• The approach is implemented in the open-source Maple package FunEq, which provides automatic proofs and handles numerous functional equations, including those from prior literature.

Experimental results

Research questions

• RQ1 Can an empirical, computer-assisted method rigorously solve algebraic-functional equations of the form $ F(P(x,t), P(x,1), x, t) \equiv 0 $ without relying on human ingenuity?
• RQ2 Is it possible to bypass the slow empirical guessing of multivariate algebraic relations by instead guessing a univariate annihilating polynomial for $ P(x,1) $?
• RQ3 How can the resultant operation be used to eliminate $ P(x,1) $ from the system and derive a valid algebraic equation for $ P(x,t) $?
• RQ4 To what extent can this method be automated and scaled to functional equations with multiple catalytic variables?
• RQ5 Can this approach provide fully rigorous proofs of known results, such as West’s conjecture on 2-stack-sortable permutations, in a fraction of the time of prior methods?

Key findings

• The method enables fully automatic and rigorous derivation of algebraic equations for bivariate generating functions $ P(x,t) $, given a known algebraic relation for $ P(x,1) $, using only empirical guessing and symbolic elimination.
• The use of the resultant to eliminate $ P(x,1) $ from the system $ F(P(x,t), Q(x), x, t) \equiv 0 $ and $ I(Q(x), x) \equiv 0 $ produces a valid algebraic equation $ G(P(x,t), x, t) \equiv 0 $, which is provably correct due to the uniqueness of formal power series solutions.
• The approach reduces the computational burden of guessing multivariate polynomials to a much faster task of guessing a bivariate polynomial in $ Q(x) $ and $ x $, significantly improving efficiency.
• The method was successfully applied to reprove the main result of Zeilberger’s earlier work on 2-stack-sortable permutations in under two seconds using the FunEq package.
• The method is fully implemented in the open-source Maple package FunEq, which provides automatic, rigorous proofs for all results in [CJS] and numerous other functional equations.
• The approach confirms that $ P $-recursive sequences and $ D $-finite generating functions naturally arise from algebraic relations, and that first-order recurrences (yielding closed forms) can be detected empirically and verified automatically.