Infinite variable polynomial

Question

I'm curious as to how to construct an infinite variable polynomial. Is there an nice formulation of such a thing? I've attempted using functions and functionals to construct one, but that didn't lead anywhere. I need it to find a separatrix for a functional-differential system of equations. I'm very appreciative of any help.

Answer 1

Well, an infinite-variable polynomial would be some kind of a function on a space like $\mathbb{R}^\omega$; and presumably it'd be a linear combination of the primitive terms $\prod_i x_i^{\alpha_i}$. You can ask yourself what further restrictions make sense. If you allow primitive terms with infinitely many nonzero exponents, then these won't be defined (i.e., the infinite product won't converge) over the whole space. That seems undesirable for a polynomial. And if you allow the linear combination to have infinitely many nonzero coefficients, then (1) the sum may not converge everywhere, and (2) even if it does, the result needn't look like a polynomial. (For instance, just over $\mathbb{R}$, $\sum_{n=0}^{\infty}x^n/n! = e^x$ isn't a polynomial.) So we can't really allow that either. You're then left with a finite sum of terms, each involving finitely many variables... in other words, an ordinary polynomial over a finite-dimensional subspace of the original space. The set of these functions is closed under addition, multiplication, and differentiation, as well as under integration with respect to a single variable; so this is probably the formulation you're looking for.