Complete Homogeneous Symmetric Polynomials with Repeating Variables
In this paper, we consider complete homogeneous symmetric polynomials evaluated for variables repeated with given multiplicities; in other words, we consider polynomials obtained from complete homogeneous polynomials by identifying some subsets of their variables. We represent such polynomials as li...
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2024-12-01
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| author | Luis Angel González-Serrano Egor A. Maximenko |
| author_facet | Luis Angel González-Serrano Egor A. Maximenko |
| author_sort | Luis Angel González-Serrano |
| collection | DOAJ |
| description | In this paper, we consider complete homogeneous symmetric polynomials evaluated for variables repeated with given multiplicities; in other words, we consider polynomials obtained from complete homogeneous polynomials by identifying some subsets of their variables. We represent such polynomials as linear combinations of the powers of the variables, where all exponents are equal to the degree of the original polynomial. We give two proofs for the proposed formulas: the first proof uses the decomposition of the generating function into partial fractions, and the second involves the inverse of the confluent Vandermonde matrix. We also discuss the computational feasibility of the proposed formulas. |
| format | Article |
| id | doaj-art-376f4cb24ebc450c8682386ba1075fd2 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-376f4cb24ebc450c8682386ba1075fd22025-01-10T13:18:01ZengMDPI AGMathematics2227-73902024-12-011313410.3390/math13010034Complete Homogeneous Symmetric Polynomials with Repeating VariablesLuis Angel González-Serrano0Egor A. Maximenko1Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Mexico City 07738, MexicoEscuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Mexico City 07738, MexicoIn this paper, we consider complete homogeneous symmetric polynomials evaluated for variables repeated with given multiplicities; in other words, we consider polynomials obtained from complete homogeneous polynomials by identifying some subsets of their variables. We represent such polynomials as linear combinations of the powers of the variables, where all exponents are equal to the degree of the original polynomial. We give two proofs for the proposed formulas: the first proof uses the decomposition of the generating function into partial fractions, and the second involves the inverse of the confluent Vandermonde matrix. We also discuss the computational feasibility of the proposed formulas.https://www.mdpi.com/2227-7390/13/1/34complete homogeneous polynomialsconfluent Vandermonde matrixpartial fractions decomposition |
| spellingShingle | Luis Angel González-Serrano Egor A. Maximenko Complete Homogeneous Symmetric Polynomials with Repeating Variables Mathematics complete homogeneous polynomials confluent Vandermonde matrix partial fractions decomposition |
| title | Complete Homogeneous Symmetric Polynomials with Repeating Variables |
| title_full | Complete Homogeneous Symmetric Polynomials with Repeating Variables |
| title_fullStr | Complete Homogeneous Symmetric Polynomials with Repeating Variables |
| title_full_unstemmed | Complete Homogeneous Symmetric Polynomials with Repeating Variables |
| title_short | Complete Homogeneous Symmetric Polynomials with Repeating Variables |
| title_sort | complete homogeneous symmetric polynomials with repeating variables |
| topic | complete homogeneous polynomials confluent Vandermonde matrix partial fractions decomposition |
| url | https://www.mdpi.com/2227-7390/13/1/34 |
| work_keys_str_mv | AT luisangelgonzalezserrano completehomogeneoussymmetricpolynomialswithrepeatingvariables AT egoramaximenko completehomogeneoussymmetricpolynomialswithrepeatingvariables |