Detecting Algebraic (In)dependence of Explicitly Presented Functions

Gurevic, Reuven Henry

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Description

Title

Detecting Algebraic (In)dependence of Explicitly Presented Functions

Author(s)

Gurevic, Reuven Henry

Issue Date

1988

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Mathematics
• Computer Science

Abstract

• I 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 ${\cal F}(U)$ the minimal field containing all functions meromorphic on $U$ and closed under exponentiation $f\mapsto e\sp{f}$. Let $f\sb{j}\in{\cal F}(U)$, $p\sb{j}\in{\cal M}(U) - \{0\}$ for 1 $\leq j \leq m$ and $g\sb{k}\in{\cal F}(U)$, $q\sb{k} \in {\cal M}(U) - \{0\}$ for $1\leq k \leq n$ (where ${\cal M}(U)$ is the field of functions meromorphic on $U).$ Let $f\sb{i} - f\sb{j} \notin {\cal H}(U)$ for $i\not= j$ and $g\sb{k} - g\sb{l} \notin{\cal H}(U)$ for $k\not= l$ (where ${\cal H}(U)$ is the ring of functions holomorphic on $U).$ If all zeros and singularities of(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$h={\sum\sbsp{j=1}{m}\ p\sb{j}e\sp{f\sb{j}} \over\sum\sbsp{k=1}{n}\ q\sb{k}e\sp{g\sb{k}}}$$(TABLE/EQUATION ENDS)are contained in an analytic subset of $U$ then $m = n$ and there exists a permutation $\sigma$ of $\{1,\...,m\}$ such that $h = (p\sb{j}/q\sb{\sigma(j)})$ $\cdot$ $e\sp{f\sb{j}-g\sb{\sigma(j)}}$ for 1 $\leq j \leq m$. When $h\in{\cal M}(U)$, additionally $f\sb{j} - g\sb{\sigma(j)}$ $\in$ ${\cal H}(U)$ for all $j$.
• On Tarski's high school algebra problem. Consider $L$ = $\{$terms in variables and 1, +, $\cdot,\uparrow\}$, where $\uparrow$: $a,b\mapsto a\sp{b}$ for positive $a,b$. Each term $t \in L$ naturally determines a function $\bar t$: (R$\sb+)\sp{n}$ $\to$ R$\sb+$, where $n$ is the number of variables involved. For $S \subset L$ put $\bar S$ = $\{\bar t\mid t \in S\}$.
• i. I describe the algebraic structure of $\bar\Lambda$ and $\bar{\cal L}$, where $\Lambda$ = $\{ t \in L \mid$ if $u\ \uparrow\ v$ occurs as a subterm of $t$ then either $u$ is a variable or $u$ contains no variables at all$\}$, and ${\cal L}$ = $\{t \in L \mid$ 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 $\bar{\cal L}$ 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 divisible neither by constants in N-$\{1\}$ nor 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 $\bar{\cal L}$ as a semiring is isomorphic to the semigroup semiring $\bar\Lambda(\oplus\sb{f}P\sb{f}),$ where each $P\sb{f}$ is a copy of $P$ and $f$ ranges over the $\ne$1 +-primes of $\bar{\cal L}$.
• ii. I prove that if $t,u \in {\cal L}$ and R$\sb+\models\ t$ = $u$ (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 (R$\sb+$; 1, +, $\cdot,$ $\uparrow)$ is not finitely axiomatizable.)

Graduation Semester

1988

Type of Resource

text

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