Sufficient conditions for a conjecture of Ryser about Hadamard Circulant matrices
Let H be a Hadamard Circulant matrix of order n=4h2 where h>1 is an odd positive integer with at least two prime divisors such that the exponents of the prime numbers that divide h are big enough and such that the nonzero coefficients of the cyclotomic polynomial Φn(t) are bounded by a constant i...
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| Veröffentlicht in: | Linear algebra and its applications 2012-12, Vol.437 (12), p.2877-2886 |
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| container_title | Linear algebra and its applications |
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| creator | Euler, Reinhardt Gallardo, Luis H. Rahavandrainy, Olivier |
| description | Let H be a Hadamard Circulant matrix of order n=4h2 where h>1 is an odd positive integer with at least two prime divisors such that the exponents of the prime numbers that divide h are big enough and such that the nonzero coefficients of the cyclotomic polynomial Φn(t) are bounded by a constant independent of n. Then for all the φ(n)n-th primitive roots w of 1, P(w)n is not an algebraic integer in the cyclotomic field K=Q(w), where P(t) is the representer polynomial of H and φ is the Euler function. This implies that P(w) is not a real number. |
| doi_str_mv | 10.1016/j.laa.2012.07.022 |
| format | Article |
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Then for all the φ(n)n-th primitive roots w of 1, P(w)n is not an algebraic integer in the cyclotomic field K=Q(w), where P(t) is the representer polynomial of H and φ is the Euler function. This implies that P(w) is not a real number.</description><identifier>ISSN: 0024-3795</identifier><identifier>EISSN: 1873-1856</identifier><identifier>DOI: 10.1016/j.laa.2012.07.022</identifier><identifier>CODEN: LAAPAW</identifier><language>eng</language><publisher>Amsterdam: Elsevier Inc</publisher><subject>Algebra ; Circulant matrices ; Computer Science ; Cyclotomic polynomials ; Discrete Mathematics ; Exact sciences and technology ; Hadamard matrices ; Kronecker’s theorem ; Linear and multilinear algebra, matrix theory ; Mathematics ; Number theory ; Primitive roots ; Ryser’s conjecture ; Sciences and techniques of general use</subject><ispartof>Linear algebra and its applications, 2012-12, Vol.437 (12), p.2877-2886</ispartof><rights>2012 Elsevier Inc.</rights><rights>2015 INIST-CNRS</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c404t-b2ec7aa0f448f6a45e04482c4a3049de795a381abc3e35e99b2d3bc0ac1f2bb93</citedby><cites>FETCH-LOGICAL-c404t-b2ec7aa0f448f6a45e04482c4a3049de795a381abc3e35e99b2d3bc0ac1f2bb93</cites><orcidid>0000-0002-4294-286X</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.laa.2012.07.022$$EHTML$$P50$$Gelsevier$$Hfree_for_read</linktohtml><link.rule.ids>230,314,780,784,885,3550,27924,27925,45995</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=26395608$$DView record in Pascal Francis$$Hfree_for_read</backlink><backlink>$$Uhttps://hal.univ-brest.fr/hal-00748316$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Euler, Reinhardt</creatorcontrib><creatorcontrib>Gallardo, Luis H.</creatorcontrib><creatorcontrib>Rahavandrainy, Olivier</creatorcontrib><title>Sufficient conditions for a conjecture of Ryser about Hadamard Circulant matrices</title><title>Linear algebra and its applications</title><description>Let H be a Hadamard Circulant matrix of order n=4h2 where h>1 is an odd positive integer with at least two prime divisors such that the exponents of the prime numbers that divide h are big enough and such that the nonzero coefficients of the cyclotomic polynomial Φn(t) are bounded by a constant independent of n. Then for all the φ(n)n-th primitive roots w of 1, P(w)n is not an algebraic integer in the cyclotomic field K=Q(w), where P(t) is the representer polynomial of H and φ is the Euler function. This implies that P(w) is not a real number.</description><subject>Algebra</subject><subject>Circulant matrices</subject><subject>Computer Science</subject><subject>Cyclotomic polynomials</subject><subject>Discrete Mathematics</subject><subject>Exact sciences and technology</subject><subject>Hadamard matrices</subject><subject>Kronecker’s theorem</subject><subject>Linear and multilinear algebra, matrix theory</subject><subject>Mathematics</subject><subject>Number theory</subject><subject>Primitive roots</subject><subject>Ryser’s conjecture</subject><subject>Sciences and techniques of general use</subject><issn>0024-3795</issn><issn>1873-1856</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2012</creationdate><recordtype>article</recordtype><recordid>eNp9kMtKxDAUhoMoOI4-gLtuXLhoPbn0hqthUEcYEG_rcJommNJph6QdmLc3pTJLVwk__5ec8xFySyGhQLOHJmkREwaUJZAnwNgZWdAi5zEt0uycLACYiHleppfkyvsGAEQObEHeP0djrLK6GyLVd7UdbN_5yPQuwilotBpGp6PeRB9Hr0Na9eMQbbDGHbo6WlunxhYDvcPBWaX9Nbkw2Hp983cuyffz09d6E2_fXl7Xq22sBIghrphWOSIYIQqToUg1hBtTAjmIstZhVOQFxUpxzVNdlhWreaUAFTWsqkq-JPfzuz_Yyr2zYZyj7NHKzWorpwwgFwWn2YGGLp27yvXeO21OAAU5-ZONDP7k5E9CLoO_wNzNzB69wtY47JT1J5BlvEwzKELvce7psOzBaif9pFPp2rogT9a9_eeXX9J8hWk</recordid><startdate>20121215</startdate><enddate>20121215</enddate><creator>Euler, Reinhardt</creator><creator>Gallardo, Luis H.</creator><creator>Rahavandrainy, Olivier</creator><general>Elsevier Inc</general><general>Elsevier</general><scope>6I.</scope><scope>AAFTH</scope><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>1XC</scope><orcidid>https://orcid.org/0000-0002-4294-286X</orcidid></search><sort><creationdate>20121215</creationdate><title>Sufficient conditions for a conjecture of Ryser about Hadamard Circulant matrices</title><author>Euler, Reinhardt ; Gallardo, Luis H. ; Rahavandrainy, Olivier</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c404t-b2ec7aa0f448f6a45e04482c4a3049de795a381abc3e35e99b2d3bc0ac1f2bb93</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2012</creationdate><topic>Algebra</topic><topic>Circulant matrices</topic><topic>Computer Science</topic><topic>Cyclotomic polynomials</topic><topic>Discrete Mathematics</topic><topic>Exact sciences and technology</topic><topic>Hadamard matrices</topic><topic>Kronecker’s theorem</topic><topic>Linear and multilinear algebra, matrix theory</topic><topic>Mathematics</topic><topic>Number theory</topic><topic>Primitive roots</topic><topic>Ryser’s conjecture</topic><topic>Sciences and techniques of general use</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Euler, Reinhardt</creatorcontrib><creatorcontrib>Gallardo, Luis H.</creatorcontrib><creatorcontrib>Rahavandrainy, Olivier</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Hyper Article en Ligne (HAL)</collection><jtitle>Linear algebra and its applications</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Euler, Reinhardt</au><au>Gallardo, Luis H.</au><au>Rahavandrainy, Olivier</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Sufficient conditions for a conjecture of Ryser about Hadamard Circulant matrices</atitle><jtitle>Linear algebra and its applications</jtitle><date>2012-12-15</date><risdate>2012</risdate><volume>437</volume><issue>12</issue><spage>2877</spage><epage>2886</epage><pages>2877-2886</pages><issn>0024-3795</issn><eissn>1873-1856</eissn><coden>LAAPAW</coden><abstract>Let H be a Hadamard Circulant matrix of order n=4h2 where h>1 is an odd positive integer with at least two prime divisors such that the exponents of the prime numbers that divide h are big enough and such that the nonzero coefficients of the cyclotomic polynomial Φn(t) are bounded by a constant independent of n. Then for all the φ(n)n-th primitive roots w of 1, P(w)n is not an algebraic integer in the cyclotomic field K=Q(w), where P(t) is the representer polynomial of H and φ is the Euler function. This implies that P(w) is not a real number.</abstract><cop>Amsterdam</cop><pub>Elsevier Inc</pub><doi>10.1016/j.laa.2012.07.022</doi><tpages>10</tpages><orcidid>https://orcid.org/0000-0002-4294-286X</orcidid><oa>free_for_read</oa></addata></record> |
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| subjects | Algebra Circulant matrices Computer Science Cyclotomic polynomials Discrete Mathematics Exact sciences and technology Hadamard matrices Kronecker’s theorem Linear and multilinear algebra, matrix theory Mathematics Number theory Primitive roots Ryser’s conjecture Sciences and techniques of general use |
| title | Sufficient conditions for a conjecture of Ryser about Hadamard Circulant matrices |
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