{"id":{"repo_id":"carleton","oai_identifier":"oai:carleton.scholaris.ca:20.500.14718/40861"},"canonical_url":"https://search.dev.ndltd.org/etd/carleton/oai:carleton.scholaris.ca:20.500.14718/40861","repository":{"repo_id":"carleton","name":"Carleton University","base_url":"https://carleton.scholaris.ca/server/oai/request"},"display":{"title":"Applications Of The Subset Sum Problem Over Finite Abelian Groups","abstract":"Given a finite abelian group $G$, a finite set $D$, and a mapping $f:D\\rightarrow G$, we find the number of $r$-subsets $S\\subseteq D$ where for $b\\in G$, \\begin{align*} \\sum_{x\\in S}f(x)=b. \\end{align*} We obtain simple exact expressions when $f$ is an abelian group homomorphism. When $G=\\Fq$, we extend known results when $D\\in\\{\\Fq,\\Fq^*\\}$ and $f(x)=x^N$, which include quadratic and semiprimitive cases. We count degree $n$ monic polynomials over $\\Fq$ with $r$ distinct roots in a set $D\\subseteq\\Fq$ when the leading terms of degree at least $n-\\ell$ are fixed. We obtain new formulas for $\\ell=1$ when $D$ is a multiplicative subgroup of $\\Fq^*$, and for $\\ell=2$ when $D$ is an arbitrary subfield of $\\Fq$ with $q$ odd.","abstract_html":"Given a finite abelian group $G$, a finite set $D$, and a mapping $f:D\\rightarrow G$, we find the number of $r$-subsets $S\\subseteq D$ where for $b\\in G$, \\begin{align*} \\sum_{x\\in S}f(x)=b. \\end{align*} We obtain simple exact expressions when $f$ is an abelian group homomorphism. When $G=\\Fq$, we extend known results when <span class=\"etd-inline-math\">D\\in\\{\\Fq,\\Fq<sup>*</sup>\\}</span> and <span class=\"etd-inline-math\">f(x)=x<sup>N</sup></span>, which include quadratic and semiprimitive cases. We count degree $n$ monic polynomials over $\\Fq$ with $r$ distinct roots in a set $D\\subseteq\\Fq$ when the leading terms of degree at least $n-\\ell$ are fixed. We obtain new formulas for $\\ell=1$ when $D$ is a multiplicative subgroup of <span class=\"etd-inline-math\">\\Fq<sup>*</sup></span>, and for $\\ell=2$ when $D$ is an arbitrary subfield of $\\Fq$ with $q$ odd.","abstract_has_math":true,"creators":["Kuttner, Simon Martial"],"institution":"Carleton University","degree_name":"Master of Science (M.Sc.)","degree_level":"Master&apos;s","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023","date_published":"2023","updated_at":"2026-07-24T01:34:41Z","subjects":[],"languages":["en"],"rights":["Copyright © 2023 the author(s). Theses may be used for non-commercial research, educational, or related academic purposes only. Such uses include personal study, distribution to students, research and scholarship. Theses may only be shared by linking to the Carleton University Institutional Repository and no part may be copied without proper attribution to the author; no part may be used for commercial purposes directly or indirectly via a for-profit platform; no adaptation or derivative works are permitted without consent from the copyright owner."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.22215/etd/2023-15572"],"render_values":[{"text":"10.22215/etd/2023-15572","href":"https://doi.org/10.22215/etd/2023-15572","code":true}]}]},"links":{"outbound_url":"https://hdl.handle.net/20.500.14718/40861","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Kuttner, Simon Martial"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-04-08T20:08:18Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2025-04-08T20:08:18Z"]},{"key":"dc:date.issued","label":"Date","values":["2023"]},{"key":"dc:publisher","label":"Institution","values":["Carleton University"]},{"key":"dc:type","label":"Dc Type","values":["thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Master&apos;s"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science (M.Sc.)"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright © 2023 the author(s). Theses may be used for non-commercial research, educational, or related academic purposes only. Such uses include personal study, distribution to students, research and scholarship. Theses may only be shared by linking to the Carleton University Institutional Repository and no part may be copied without proper attribution to the author; no part may be used for commercial purposes directly or indirectly via a for-profit platform; no adaptation or derivative works are permitted without consent from the copyright owner."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.22215/etd/2023-15572"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/20.500.14718/40861"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Given a finite abelian group $G$, a finite set $D$, and a mapping $f:D\\rightarrow G$, we find the number of $r$-subsets $S\\subseteq D$ where for $b\\in G$, \\begin{align*} \\sum_{x\\in S}f(x)=b. \\end{align*} We obtain simple exact expressions when $f$ is an abelian group homomorphism. When $G=\\Fq$, we extend known results when $D\\in\\{\\Fq,\\Fq^*\\}$ and $f(x)=x^N$, which include quadratic and semiprimitive cases. We count degree $n$ monic polynomials over $\\Fq$ with $r$ distinct roots in a set $D\\subseteq\\Fq$ when the leading terms of degree at least $n-\\ell$ are fixed. We obtain new formulas for $\\ell=1$ when $D$ is a multiplicative subgroup of $\\Fq^*$, and for $\\ell=2$ when $D$ is an arbitrary subfield of $\\Fq$ with $q$ odd."]},{"key":"dc:title","label":"Title","values":["Applications Of The Subset Sum Problem Over Finite Abelian Groups"]}]}],"canonical_facts":{"dc:creator":["Kuttner, Simon Martial"],"dc:date.accessioned":["2025-04-08T20:08:18Z"],"dc:date.available":["2025-04-08T20:08:18Z"],"dc:date.issued":["2023"],"dc:description.abstract":["Given a finite abelian group $G$, a finite set $D$, and a mapping $f:D\\rightarrow G$, we find the number of $r$-subsets $S\\subseteq D$ where for $b\\in G$, \\begin{align*} \\sum_{x\\in S}f(x)=b. \\end{align*} We obtain simple exact expressions when $f$ is an abelian group homomorphism. When $G=\\Fq$, we extend known results when $D\\in\\{\\Fq,\\Fq^*\\}$ and $f(x)=x^N$, which include quadratic and semiprimitive cases. We count degree $n$ monic polynomials over $\\Fq$ with $r$ distinct roots in a set $D\\subseteq\\Fq$ when the leading terms of degree at least $n-\\ell$ are fixed. We obtain new formulas for $\\ell=1$ when $D$ is a multiplicative subgroup of $\\Fq^*$, and for $\\ell=2$ when $D$ is an arbitrary subfield of $\\Fq$ with $q$ odd."],"dc:identifier.doi":["10.22215/etd/2023-15572"],"dc:identifier.uri":["https://hdl.handle.net/20.500.14718/40861"],"dc:language.iso":["en"],"dc:publisher":["Carleton University"],"dc:rights":["Copyright © 2023 the author(s). Theses may be used for non-commercial research, educational, or related academic purposes only. Such uses include personal study, distribution to students, research and scholarship. Theses may only be shared by linking to the Carleton University Institutional Repository and no part may be copied without proper attribution to the author; no part may be used for commercial purposes directly or indirectly via a for-profit platform; no adaptation or derivative works are permitted without consent from the copyright owner."],"dc:title":["Applications Of The Subset Sum Problem Over Finite Abelian Groups"],"dc:type":["thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Master&apos;s"],"thesis:degree_name":["Master of Science (M.Sc.)"]},"updated_at":"2026-07-24T01:34:41Z"}