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 L1 and L2 of L are definably equivalent if each relation in L1 can be defined by an L2-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 Lm denote the language of multiplication alone, and let La 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 Lf denote the language consisting of the relation defined by f. The possibilities for Lm ∨ Lf are examined in §2, and the possibilities for La ∨ Lf 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 Lm and that the only languages between La and L are expansions of La by scalar multiplications.
Canonical forms of 2×3×3 tensors over the real field, algebraically closed fields, and finite fields