Detecting algebraic (in)dependence of explicitly presented functions (some applications of Nevanlinna theory to mathematical logic)

Author:
R. H. Gurevič

Journal:
Trans. Amer. Math. Soc. **336** (1993), 1-67

MSC:
Primary 03C62; Secondary 03B30, 30D35, 32A22

DOI:
https://doi.org/10.1090/S0002-9947-1993-0991961-1

MathSciNet review:
991961

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Abstract: We consider algebraic relations between explicitly presented analytic functions with particular emphasis on Tarski's high school algebra problem.

*The part not related directly to Tarski's high school algebra problem*. Let be a connected complex-analytic manifold. Denote by the minimal field containing all functions meromorphic on and closed under exponentiation . Let , for , and , for (where is the field of functions meromorphic on ). Let for and for (where is the ring of functions holomorphic on ). If all zeros and singularities of

*On Tarski's high school algebra problem*. Consider -terms in variables and , , , , where , for positive , . Each term naturally determines a function : , where is the number of variables involved. For put .

(i) We describe the algebraic structure of and , where if occurs as a subterm of then either is a variable or contains no variables at all, and if occurs as a subterm of then . Of these, is a free semiring with respect to addition and multiplication but is free only as a semigroup with respect to addition. A function is called -prime in if for all , and is called multiplicatively prime in if or for , . A function is called -prime in if it is both -prime and multiplicatively prime in . A function in is said to have content if it is not divisible by constants in or by -primes of . The product of functions of content has content . Let be the multiplicative subsemigroup of of functions of content . Then as a semiring is isomorphic to the semigroup semiring , where each is a copy of and ranges over the -primes of .

(ii) We prove that if , and (i.e., if ) then Tarski's "high school algebra" identities . This result covers a conjecture of C. W. Henson and L. A. Rubel. (Note: this result does not generalize to arbitrary , . Moreover, the equational theory of is not finitely axiomatizable.

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DOI:
https://doi.org/10.1090/S0002-9947-1993-0991961-1

Article copyright:
© Copyright 1993
American Mathematical Society