Combinatorial Factorization Theory

Many algebraic/mathematical objects (mathematicians speak of elements) allow factorizations (decompositions) into simpler objects, where simpler objects are of the same type but allow no further decompositions. Such simpler objects are called irreducible or indecomposable or atoms (as it is the case in physics). We study factorizations of given objects in order to get a better understanding of the structure (nature) of the initial objects. A simple example, we can think of, are the positive integers. In this case the simpler objects are the prime numbers. Prime numbers are positive integers which allow no further factorizations into positive integers strictly greater than one, and every positive integer allows a factorization into prime numbers. In the same way, polynomials with integer (or with real or complex) coecients allow a factorization into irreducible polynomials. Positive integers have precisely one factorization into prime numbers. In general, algebraic objects allow a large variety of factorizations into irreducibles (in other words, the big objects can be glued together by atoms in many dierent ways). Let us consider polynomials with non- negative integer coecients. They can be factored into irreducible polynomials with non-negative coecients but in general there are many distinct factorizations of this type. Factorization theory studies factorizations of objects into irreducible ones. Its goal is to describe all the distinct factorizations of one xed elements, in other words to classify the non-uniqueness of factorizations by suitable algebraic parameters, such as sets of lengths (if an element a is a product of k irreducibles, then k is called a factorization length, and all such factorization lengths together form the set of lengths of a; thus a set of lengths is a set of positive integers). Combinatorial Factorization Theory studies the non-uniqueness of factorizations of elements in Krull monoids with (discrete) methods from Additive Combinatorics. Krull monoids are suitable subsets (subareas) of sets where factorizations are unique, and they comprise many generalized polynomial rings. Discrete mathematics oftentimes studies a nite variety of objects (knots in a nite network, algorithms, words in a programming language), whence it has close connections to computer science.