Definability in reducts of algebraically closed fields

Extract

Let K be an algebraically closed field and let L be its canonical language; that is, L consists of all relations on K which are definable from addition, multiplication, and parameters from K. Two sublanguages L ₁ and L ₂ of L are definably equivalent if each relation in L ₁ can be defined by an L ₂-formula with parameters in K, and vice versa. The equivalence classes of sublanguages of L form a quotient lattice of the power set of L about which very little is known. We will not distinguish between a sublanguage and its equivalence class.

Let L _m denote the language of multiplication alone, and let L _a denote the language of addition alone. Let f ∈ K [X, Y] and consider the algebraic function defined by f (x, y) = 0 for x, y ∈ K. Let L _f denote the language consisting of the relation defined by f. The possibilities for L _m ∨ L _f are examined in §2, and the possibilities for L _a ∨ L _f are examined in §3. In fact the only comprehensive results known are under the additional hypothesis that f actually defines a rational function (i.e., when f is linear in one of the variables), and in positive characteristic, only expansions of addition by polynomials (i.e., when f is linear and monic in one of the variables) are understood. It is hoped that these hypotheses will turn out to be unnecessary, so that reasonable generalizations of the theorems described below to algebraic functions will be true. The conjecture is that L covers L _m and that the only languages between L _a and L are expansions of L _a by scalar multiplications.