ISSN 1088-6850(online) ISSN 0002-9947(print)

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

are contained in an analytic subset of then and there exists a permutation of such that for . When , additionally for all .

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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