Linear dependence of quotients of analytic functions of several variables with the least subcollection of generalized Wronskians
We study linear dependence in the case of quotients of analytic functions in several variables (real or complex). We identify the least subcollection of generalized Wronskians whose identical vanishing is sufficient for linear dependence. Our proof admits a straight-forward algebraic generalization...
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| Published in: | Linear algebra and its applications Vol. 408; pp. 151 - 160 |
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| Format: | Journal Article |
| Language: | English |
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New York, NY
Elsevier Inc
01.10.2005
Elsevier Science |
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| ISSN: | 0024-3795, 1873-1856 |
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| Abstract | We study linear dependence in the case of quotients of analytic functions in several variables (real or complex). We identify the least subcollection of generalized Wronskians whose identical vanishing is sufficient for linear dependence. Our proof admits a straight-forward algebraic generalization and also constitutes an alternative proof of the previously known result that the identical vanishing of the whole collection of generalized Wronskians implies linear dependence. Motivated by the structure of this proof, we introduce a method for calculating the space of linear relations. We conclude with some reflections about this method that may be promising from a computational point of view. |
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| AbstractList | We study linear dependence in the case of quotients of analytic functions in several variables (real or complex). We identify the least subcollection of generalized Wronskians whose identical vanishing is sufficient for linear dependence. Our proof admits a straight-forward algebraic generalization and also constitutes an alternative proof of the previously known result that the identical vanishing of the whole collection of generalized Wronskians implies linear dependence. Motivated by the structure of this proof, we introduce a method for calculating the space of linear relations. We conclude with some reflections about this method that may be promising from a computational point of view. |
| Author | Walker, Ronald A. |
| Author_xml | – sequence: 1 givenname: Ronald A. surname: Walker fullname: Walker, Ronald A. email: rawalker@math.ohio-state.edu organization: Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, OH 43210-1174, USA |
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| Cites_doi | 10.2307/2007186 10.1112/S0025579300000644 10.1016/0024-3795(89)90548-X 10.1007/BF01458694 10.2307/2311681 10.1080/17476939408814706 10.2307/1986214 10.1016/0024-3795(89)90393-5 10.1007/BF01450000 |
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| Keywords | Higher-dimensional partitions Linear dependence 32A20 15A54 Young-like sets Generalized Wronskians 15A03 Analytic functions of several variables 32A20 Linear dependence Young- like sets Linear relation Multivariate analysis Generalized Wronskian Quotient Analytical function |
| Language | English |
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| References | Meisters (bib8) 1961; 68 Bôcher (bib4) 1901; 2 Peano (bib10) 1897 Wolsson (bib13) 1989; 116 Roth (bib11) 1955; 2 Kellogg (bib7) 1912; 41 Bôcher (bib3) 1900; 2 Cassels (bib5) 1957 Siegel (bib12) 1921; 84 Berenstein, Chang, Li (bib2) 1994; 24 Curtiss (bib6) 1908; 65 Wolsson (bib14) 1989; 117 Andrews (bib1) 1998 Peano (bib9) 1889; 9 Cassels (10.1016/j.laa.2005.06.002_bib5) 1957 Curtiss (10.1016/j.laa.2005.06.002_bib6) 1908; 65 Wolsson (10.1016/j.laa.2005.06.002_bib13) 1989; 116 Siegel (10.1016/j.laa.2005.06.002_bib12) 1921; 84 Berenstein (10.1016/j.laa.2005.06.002_bib2) 1994; 24 Wolsson (10.1016/j.laa.2005.06.002_bib14) 1989; 117 Meisters (10.1016/j.laa.2005.06.002_bib8) 1961; 68 Bôcher (10.1016/j.laa.2005.06.002_bib4) 1901; 2 Roth (10.1016/j.laa.2005.06.002_bib11) 1955; 2 Peano (10.1016/j.laa.2005.06.002_bib9) 1889; 9 Andrews (10.1016/j.laa.2005.06.002_bib1) 1998 Bôcher (10.1016/j.laa.2005.06.002_bib3) 1900; 2 Kellogg (10.1016/j.laa.2005.06.002_bib7) 1912; 41 Peano (10.1016/j.laa.2005.06.002_bib10) 1897 |
| References_xml | – volume: 24 start-page: 131 year: 1994 end-page: 144 ident: bib2 article-title: A note on Wronskians and linear dependence of entire functions in publication-title: Complex Variables Theory Appl. – volume: 2 start-page: 139 year: 1901 end-page: 149 ident: bib4 article-title: Certain cases in which the vanishing of the Wronskian is a sufficient condition for linear dependence publication-title: Trans. Amer. Math. Soc. – volume: 65 start-page: 282 year: 1908 end-page: 298 ident: bib6 article-title: The vanishing of the Wronskian and the problem of linear dependence publication-title: Math. Ann. – volume: 9 start-page: 75 year: 1889 end-page: 76 ident: bib9 article-title: Sur le determinant Wronskien publication-title: Mathesis – volume: 116 start-page: 1 year: 1989 end-page: 8 ident: bib13 article-title: A condition equivalent to linear dependence for functions with vanishing Wronskian publication-title: Linear Algebra Appl. – volume: 41 start-page: 19 year: 1912 end-page: 21 ident: bib7 publication-title: C.R. Soc. Math. France – volume: 2 start-page: 81 year: 1900 end-page: 96 ident: bib3 article-title: The theory of linear dependence publication-title: Ann. Math. – year: 1957 ident: bib5 article-title: An Introduction to Diophantine Approximation – volume: 117 start-page: 73 year: 1989 end-page: 80 ident: bib14 article-title: Linear dependence of a function set of publication-title: Linear Algebra Appl. – volume: 68 start-page: 847 year: 1961 end-page: 856 ident: bib8 article-title: Local linear dependence and the vanishing of the Wronskian publication-title: Amer. Math. Monthly – volume: 84 start-page: 80 year: 1921 end-page: 99 ident: bib12 article-title: Über Näherungswerte algbraischer Zahlen publication-title: Math. Ann. – year: 1998 ident: bib1 article-title: The Theory of Partitions – start-page: 413 year: 1897 end-page: 415 ident: bib10 article-title: Sul determinante Wronskiano publication-title: Atti Accad. Naz. Lincei Rendi. Cl. Sci. Fis. Mat. Natur. – volume: 2 start-page: 1 year: 1955 end-page: 20 ident: bib11 article-title: Rational approximations to algebraic numbers publication-title: Mathematika – volume: 2 start-page: 81 year: 1900 ident: 10.1016/j.laa.2005.06.002_bib3 article-title: The theory of linear dependence publication-title: Ann. Math. doi: 10.2307/2007186 – volume: 2 start-page: 1 year: 1955 ident: 10.1016/j.laa.2005.06.002_bib11 article-title: Rational approximations to algebraic numbers publication-title: Mathematika doi: 10.1112/S0025579300000644 – start-page: 413 issue: 5 year: 1897 ident: 10.1016/j.laa.2005.06.002_bib10 article-title: Sul determinante Wronskiano publication-title: Atti Accad. Naz. Lincei Rendi. Cl. Sci. Fis. Mat. Natur. – volume: 117 start-page: 73 year: 1989 ident: 10.1016/j.laa.2005.06.002_bib14 article-title: Linear dependence of a function set of m variables with vanishing generalized Wronskians publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(89)90548-X – year: 1957 ident: 10.1016/j.laa.2005.06.002_bib5 – volume: 84 start-page: 80 year: 1921 ident: 10.1016/j.laa.2005.06.002_bib12 article-title: Über Näherungswerte algbraischer Zahlen publication-title: Math. Ann. doi: 10.1007/BF01458694 – volume: 41 start-page: 19 year: 1912 ident: 10.1016/j.laa.2005.06.002_bib7 publication-title: C.R. Soc. Math. France – volume: 68 start-page: 847 year: 1961 ident: 10.1016/j.laa.2005.06.002_bib8 article-title: Local linear dependence and the vanishing of the Wronskian publication-title: Amer. Math. Monthly doi: 10.2307/2311681 – volume: 24 start-page: 131 year: 1994 ident: 10.1016/j.laa.2005.06.002_bib2 article-title: A note on Wronskians and linear dependence of entire functions in Cn publication-title: Complex Variables Theory Appl. doi: 10.1080/17476939408814706 – volume: 2 start-page: 139 year: 1901 ident: 10.1016/j.laa.2005.06.002_bib4 article-title: Certain cases in which the vanishing of the Wronskian is a sufficient condition for linear dependence publication-title: Trans. Amer. Math. Soc. doi: 10.2307/1986214 – volume: 9 start-page: 75 year: 1889 ident: 10.1016/j.laa.2005.06.002_bib9 article-title: Sur le determinant Wronskien publication-title: Mathesis – volume: 116 start-page: 1 year: 1989 ident: 10.1016/j.laa.2005.06.002_bib13 article-title: A condition equivalent to linear dependence for functions with vanishing Wronskian publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(89)90393-5 – year: 1998 ident: 10.1016/j.laa.2005.06.002_bib1 – volume: 65 start-page: 282 year: 1908 ident: 10.1016/j.laa.2005.06.002_bib6 article-title: The vanishing of the Wronskian and the problem of linear dependence publication-title: Math. Ann. doi: 10.1007/BF01450000 |
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| Snippet | We study linear dependence in the case of quotients of analytic functions in several variables (real or complex). We identify the least subcollection of... |
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| SubjectTerms | Algebra Analytic functions of several variables Exact sciences and technology Generalized Wronskians Higher-dimensional partitions Linear and multilinear algebra, matrix theory Linear dependence Mathematics Sciences and techniques of general use Young-like sets |
| Title | Linear dependence of quotients of analytic functions of several variables with the least subcollection of generalized Wronskians |
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