Perlen-Kolloquium: Giulio Caviglia (Purdue University) and Olga Dmitrieva (Stanford University)

Monomial orders: from computational algebra to linguistic typology

A monomial order is a total order, compatible with multiplication, defined on the set of all monomials of a polynomial ring. In 1920's Macaulay introduced these orderings to characterize all possible Hilbert functions of graded ideals by comparing them to monomial ideals. Subsequently Gröbner and his student Bruno Buchberger used them to associate to a graded ideal a set of multivariate polynomials with desirable algorithmic properties, called a Gröbner basis. Gröbner bases have a wide range of applications, not only in algebra and algebraic geometry but also in many sciences in which polynomial models are used. A general principle is that many numerical invariants of a set of polynomials can be computed, or at least bounded, by studying certain monomials determined by a Gröbner basis.

In this lecture I will first give an overview of this principle in both commutative and computational algebra. I will then use monomial orders to explain the analogies between two distinct, yet similar, theories in phonology: Optimality Theory and Harmonic Grammars. In particular I will show how to use a mathematical object, called Gröbner fan, to build models of linguistic typology. 

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