Column symmetric polynomials
Eduardo Dubuc, Anders Kock

TL;DR
This paper investigates a specialized polynomial algebra with column-based relations, revealing its structure as a quotient of a simpler algebra and exploring applications in synthetic differential geometry.
Contribution
It characterizes the invariant sub-algebra under column permutations as a quotient of a polynomial algebra in fewer variables.
Findings
Invariant sub-algebra is a quotient of polynomial algebra in m variables
The quotient map relates variables to row sums of the matrix
Application to synthetic differential geometry is proposed
Abstract
We study the polynomial algebra (over a ring containing the rationals) in an n by m matrix of variables, and subject to the relation that says that the product of any two variables in the same column is zero. We show that the sub-algebra of polynomials, which are invariant under n! permutations of the columns, is a quotient of the polynomial algebra in m variables; the quotient map sends the ith variable to the sum of the entries in the ith row of the matrix. - An application in synthetic differential geometry is sketched
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Polynomial and algebraic computation · Nonlinear Waves and Solitons
