Peak Reduction Technique in Commutative Algebra

The “peak reduction” method is a powerful combinatorial technique with applications in many different areas of mathematics as well as theoretical computer science. It was originally created by Whitehead, a famous topologist and group theorist, who used it to solve an important algorithmic problem concerning automorphisms of a free group. Since then, this method was used to solve numerous problems in group theory, topology, combinatorics, and probably in some other areas as well. In this paper, we present what seems to be the first applications of this technique in commutative algebra and affine algebraic geometry. We contribute toward a classification of two-variable polynomials by classifying, up to an automorphism, polynomials of the form $$a{\chi ^n} + b{y^m} + \sum\nolimits_{im + jn \leqslant mn} {cij{\chi ^i}} {y^j}$$ (i.e., polynomials whose Newton polygon is either a triangle or a line segment). This has several applications to the study of embeddings of algebraic curves in the plane. In particular, upon combining our method with a well-known theorem of Zaidenberg and Lin, we show that one can decide “almost” just by inspection whether or not a polynomial fiber {p(x, y) = 0} is an irreducible simply connected curve. Another application that we present here, yields a new decomposition of the group Aut(K(x, y)) in a free product with amalgamation.