Polynomials as terms and the Boolean Independence Theorem

Abstract

We develop a theory of formal multivariate polynomials over commutative rings by treating them as ring terms. Our main result is that two ring terms are s-equivalent (when expanded they yield the same standard polynomial) iff they are f-equivalent (one can be transformed in the other by a series of elementary transformations). We consider in a similar way Boolean terms (formulas) and prove a theorem that two events $a$ and $b$ in a probability space, which are built by two Boolean terms from respective tuples $A$ and $B$ of elementary events, are independent if the events in $A$ are independent of the events in $B$. This theorem rigorizes arguments in the Probabilistic Method in Combinatorics.