A description of two­generated subalgebras of a polynomial ring in one variable and a new proof of the AMS theorem

Conferência de Álgebra

Leonid Makar­Limanov (Wayne University)

Abstract: The famous AMS (Abhyankar­Moh­Suzuki) theorem states that if two polynomials f and g in one variable with coefficients in a field F generate all algebra of polynomials, i.e. any polynomial h in one variable can be expressed as h = H(f, g) where H is a polynomial in two  variables, then either the degree of f divides the degree of g, or the degree of g divides the degree of f, or the degree of f and the degree of g are divisible by the characteristic of the field F. There were several wrong published proofs of this theorem and there are many correct published proofs of this theorem but all of them either long or not self­contained. Recently I found a (relatively) short and self­contained proof which will be discussed. The talk is accessible to undergraduate students knowing elementary linear algebra.