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 introduced 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 give a survey of what seems to be the first applications of the peak reduction technique in commutative algebra and affine algebraic geometry. Using this technique, we have contributed toward a classification of two-variable polynomials having classified, up to an automorphism, polynomials of the form ax+by+ ∑ im+jn≤mn cijx y (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 have shown that one can decide “almost” just by inspection whether or not a polynomial fiber {p(x, y) = 0} is an irreducible simply connected curve in C. Recently, P.Wightwick used the idea of peak reduction in combination with splice diagrams technique due to D.Eisenbud and W.Neumann to classify all twovariable polynomials over C up to an automorphism. Another application that we present here, yields a decomposition of the group Aut(K[x, y]) in a free product with amalgamation.