{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71270"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71270","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Detecting Algebraic (In)dependence of Explicitly Presented Functions","abstract":"I consider algebraic relations between explicitly presented analytic functions with particular emphasis on Tarski's high school algebra problem.","abstract_html":"I consider algebraic relations between explicitly presented analytic functions with particular emphasis on Tarski's high school algebra problem.","abstract_has_math":false,"creators":["Gurevic, Reuven Henry"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:27Z","date_published":"2014-12-16T06:18:27Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics","Computer Science"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8908693"],"render_values":[{"text":"(UMI)AAI8908693","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71270","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Gurevic, Reuven Henry"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:27Z","10000-01-01","1988"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Computer Science"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71270","(UMI)AAI8908693"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.)","Made available in DSpace on 2014-12-16T06:18:27Z (GMT). No. of bitstreams: 1 8908693.pdf: 4523302 bytes, checksum: 64e32310da9dcc10409a317589162f71 (MD5) Previous issue date: 1988","Embargo set by: Seth Robbins for item 71436 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","105 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988."]},{"key":"dc:title","label":"Title","values":["Detecting Algebraic (In)dependence of Explicitly Presented Functions"]}]}],"canonical_facts":{"dc:creator":["Gurevic, Reuven Henry"],"dc:date":["2014-12-16T06:18:27Z","10000-01-01","1988"],"dc:description":["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.)","Made available in DSpace on 2014-12-16T06:18:27Z (GMT). No. of bitstreams: 1 8908693.pdf: 4523302 bytes, checksum: 64e32310da9dcc10409a317589162f71 (MD5) Previous issue date: 1988","Embargo set by: Seth Robbins for item 71436 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","105 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988."],"dc:identifier":["http://hdl.handle.net/2142/71270","(UMI)AAI8908693"],"dc:subject":["Mathematics","Computer Science"],"dc:title":["Detecting Algebraic (In)dependence of Explicitly Presented Functions"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:04Z"}