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Polynomials as terms and the Boolean Independence Theorem

M. Klazar

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.

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 and in a probability space, which are built by two Boolean terms from respective tuples and of elementary events, are independent if the events in are independent of the events in . This theorem rigorizes arguments in the Probabilistic Method in Combinatorics.
Paper Structure (10 sections, 34 theorems, 184 equations)

This paper contains 10 sections, 34 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Notation and notions
  3. Standard polynomials
  4. Terms and evaluation maps
  5. s-equivalence and f-equivalence of ring terms
  6. Boolean algebras and Boolean formulas
  7. DNF terms
  8. The Boolean Independence Theorem
  9. The BIT in the Probabilistic Method
  10. Concluding remarks

Key Result

Theorem 3.5

Let $n\in\mathbb{N}$, $R$ be a ring and $\mathcal{R}=R[x_1,\dots,x_n]$, resp. $\mathcal{S}=R[x_1,x_2,\dots]$. Then is a ring.

Theorems & Definitions (58)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Theorem 3.5
  • Definition 3.6
  • Definition 4.1
  • Theorem 4.2
  • Corollary 4.3
  • Definition 4.4
  • ...and 48 more