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

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 $U$ be a connected complex-analytic manifold. Denote by $\mathcal {F}(U)$ the minimal field containing all functions meromorphic on $U$ and closed under exponentiation $f \mapsto {e^f}$. Let ${f_j} \in \mathcal {F}(U)$, ${p_j} \in \mathcal {M}(U) - \{ 0\}$ for $1 \leq j \leq m$, and ${g_k} \in \mathcal {F}(U)$, ${q_k} \in \mathcal {M}(U) - \{ 0\}$ for $1 \leq k \leq n$ (where $\mathcal {M}(U)$ is the field of functions meromorphic on $U$). Let ${f_i} - {f_j} \notin \mathcal {H}(U)$ for $i \ne j$ and ${g_k} - {g_l} \notin \mathcal {H}(U)$ for $k \ne 1$ (where $\mathcal {H}(U)$ is the ring of functions holomorphic on $U$). If all zeros and singularities of \[ h = \frac {{\sum \nolimits _{j = 1}^m {{p_j}{e^{{f_j}}}} }} {{\sum \nolimits _{k = 1}^n {{q_k}{e^{{g_k}}}} }}\] are contained in an analytic subset of $U$ then $m = n$ and there exists a permutation $\sigma$ of $\{ 1, \ldots ,m\}$ such that $h = ({p_j}/{q_{\sigma (j)}}) \cdot {e^{{f_j} - {g_{\sigma (j)}}}}$ for $1 \leq j \leq m$. When $h \in \mathcal {M}(U)$, additionally ${f_j} - {g_{\sigma (j)}} \in \mathcal {H}(U)$ for all $j$ . On Tarski’s high school algebra problem. Consider $L = \{$-terms in variables and $1$, $+$, $\cdot$, $\uparrow \}$ , where $\uparrow :a$, $b \mapsto {a^b}$ for positive $a$, $b$. Each term $t \in L$ naturally determines a function $\bar t$ : ${({{\mathbf {R}}_ + })^n} \to {{\mathbf {R}}_ - }$ , where $n$ is the number of variables involved. For $S \subset L$ put $\bar S = \{ \bar t|t \in S\}$ . (i) We describe the algebraic structure of $\bar \Lambda$ and $\bar {\mathcal {L}}$ , where $\Lambda = \{ t \in L|$ if $u \uparrow v$ occurs as a subterm of $t$ then either $u$ is a variable or $u$ contains no variables at all} , and $\mathcal {L} = \{ t \in L|$ if $u \uparrow v$ occurs as a subterm of $t$ then $u \in \Lambda \}$. Of these, $\bar \Lambda$ is a free semiring with respect to addition and multiplication but $\bar {\mathcal {L}}$ is free only as a semigroup with respect to addition. A function $\bar t \in \bar S$ is called $+$-prime in $\bar S$ if $\bar t \ne \bar u + \bar v$ for all $u$, $v \in S$ and is called multiplicatively prime in $\bar S$ if $\bar t = \bar u \cdot \bar v \Rightarrow \bar u = 1$ or $\bar v = 1$ for $u$, $v \in S$. A function is called $( + , \cdot )$-prime in $\bar S$ if it is both $+$-prime and multiplicatively prime in $\bar S$. A function in $\bar \Lambda$ is said to have content $1$ if it is not divisible by constants in ${\mathbf {N}} - \{ 1\}$ or by $\ne 1\ ( + , \cdot )$-primes of $\bar \Lambda$ . The product of functions of content $1$ has content $1$ . Let $P$ be the multiplicative subsemigroup of $\bar \Lambda$ of functions of content $1$ . Then $\bar {\mathcal {L}}$ as a semiring is isomorphic to the semigroup semiring $\bar \Lambda ({ \oplus _f}{P_f})$, where each ${P_f}$ is a copy of $P$ and $f$ ranges over the $\ne 1\; +$-primes of $\bar {\mathcal {L}}$. (ii) We prove that if $t$, $u \in \mathcal {L}$ and ${{\mathbf {R}}_ + } \vdash t = u$ (i.e., if $\bar t = \bar u$) then Tarski’s "high school algebra" identities $\vdash t = u$. This result covers a conjecture of C. W. Henson and L. A. Rubel. (Note: this result does not generalize to arbitrary $t$, $u \in L$ . Moreover, the equational theory of $({{\mathbf {R}}_ + };\;1, + , \cdot , \uparrow )$ is not finitely axiomatizable.