Eilenberg–Niven theorem: Difference between revisions
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→Statement: P(x) → P(x) = 0, since equations have solutions, not polynomials. →Generalizations: Add “counterexample”. |
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: <math>P(x) = a_0 x a_1 x \cdots x a_n + \varphi(x)</math> |
: <math>P(x) = a_0 x a_1 x \cdots x a_n + \varphi(x)</math> |
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where ''x'', ''a''<sub>''0''</sub>, ''a''<sub>''1''</sub>, ... , ''a''<sub>''n''</sub> are non-zero quaternions and ''φ''(''x'') is a finite sum of monomials similar to the first term but with degree less than ''n''. Then ''P''(''x'') has at least one solution.<ref>{{Cite journal|last1=Eilenberg|first1=Samuel|last2=Niven|first2=Ivan|date=April 1944|title=The "fundamental theorem of algebra" for quaternions|url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-50/issue-4/The-fundamental-theorem-of-algebra-for-quaternions/bams/1183505735.full|journal=[[Bulletin of the American Mathematical Society]]|volume=50|issue=4|pages=246–248|doi=10.1090/S0002-9904-1944-08125-1 |doi-access=free}}</ref> |
where ''x'', ''a''<sub>''0''</sub>, ''a''<sub>''1''</sub>, ... , ''a''<sub>''n''</sub> are non-zero quaternions and ''φ''(''x'') is a finite sum of monomials similar to the first term but with degree less than ''n''. Then ''P''(''x'') = 0 has at least one solution.<ref>{{Cite journal|last1=Eilenberg|first1=Samuel|last2=Niven|first2=Ivan|date=April 1944|title=The "fundamental theorem of algebra" for quaternions|url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-50/issue-4/The-fundamental-theorem-of-algebra-for-quaternions/bams/1183505735.full|journal=[[Bulletin of the American Mathematical Society]]|volume=50|issue=4|pages=246–248|doi=10.1090/S0002-9904-1944-08125-1 |doi-access=free}}</ref> |
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== Generalizations == |
== Generalizations == |
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If permitting multiple monomials with the highest degree, then the theorem does not hold, and ''P''(''x'') = ''x'' + '''i'''''x'''''i''' + 1 = 0 is a counterexample with no solutions. |
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Eilenberg–Niven theorem can also be generalized to [[octonion]]s: all octonionic polynomials with a unique monomial of higher degree have at least one solution, independent of the order of the parenthesis (the octonions are a [[non-associative algebra]]).<ref>{{Cite journal|last1=Liu|first1=Ming-Sheng|last2=Xiang|first2=Na|last3=Yang|first3=Yan|date=2017|title=On the Zeroes of Clifford Algebra-Valued Polynomials with Paravector Coefficients|url=http://link.springer.com/10.1007/s00006-016-0748-9|journal=Advances in Applied Clifford Algebras|language=en|volume=27|issue=2|pages=1531–1550|doi=10.1007/s00006-016-0748-9|s2cid=253598676 |issn=0188-7009}}</ref><ref>{{Cite journal|last=Jou|first=Yuh-Lin|date=1950|title=The "fundamental theorem of algebra" for Cayley numbers|journal=Acad. Sinica Science Record|volume=3|pages=29–33}}</ref> Different from quaternions, however, the monic and non-monic octonionic polynomials do not have always the same set of zeros.<ref>{{Cite journal|last=Serôdio|first=Rogério|date=2007|title=On Octonionic Polynomials|url=http://link.springer.com/10.1007/s00006-007-0026-y|journal=Advances in Applied Clifford Algebras|language=en|volume=17|issue=2|pages=245–258|doi=10.1007/s00006-007-0026-y|s2cid=123578310 |issn=0188-7009}}</ref> |
Eilenberg–Niven theorem can also be generalized to [[octonion]]s: all octonionic polynomials with a unique monomial of higher degree have at least one solution, independent of the order of the parenthesis (the octonions are a [[non-associative algebra]]).<ref>{{Cite journal|last1=Liu|first1=Ming-Sheng|last2=Xiang|first2=Na|last3=Yang|first3=Yan|date=2017|title=On the Zeroes of Clifford Algebra-Valued Polynomials with Paravector Coefficients|url=http://link.springer.com/10.1007/s00006-016-0748-9|journal=Advances in Applied Clifford Algebras|language=en|volume=27|issue=2|pages=1531–1550|doi=10.1007/s00006-016-0748-9|s2cid=253598676 |issn=0188-7009}}</ref><ref>{{Cite journal|last=Jou|first=Yuh-Lin|date=1950|title=The "fundamental theorem of algebra" for Cayley numbers|journal=Acad. Sinica Science Record|volume=3|pages=29–33}}</ref> Different from quaternions, however, the monic and non-monic octonionic polynomials do not have always the same set of zeros.<ref>{{Cite journal|last=Serôdio|first=Rogério|date=2007|title=On Octonionic Polynomials|url=http://link.springer.com/10.1007/s00006-007-0026-y|journal=Advances in Applied Clifford Algebras|language=en|volume=17|issue=2|pages=245–258|doi=10.1007/s00006-007-0026-y|s2cid=123578310 |issn=0188-7009}}</ref> |
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Latest revision as of 07:16, 15 August 2023
The Eilenberg–Niven theorem is a theorem that generalizes the fundamental theorem of algebra to quaternionic polynomials, that is, polynomials with quaternion coefficients and variables. It is due to Samuel Eilenberg and Ivan M. Niven.
Statement
[edit]Let
where x, a0, a1, ... , an are non-zero quaternions and φ(x) is a finite sum of monomials similar to the first term but with degree less than n. Then P(x) = 0 has at least one solution.[1]
Generalizations
[edit]If permitting multiple monomials with the highest degree, then the theorem does not hold, and P(x) = x + ixi + 1 = 0 is a counterexample with no solutions.
Eilenberg–Niven theorem can also be generalized to octonions: all octonionic polynomials with a unique monomial of higher degree have at least one solution, independent of the order of the parenthesis (the octonions are a non-associative algebra).[2][3] Different from quaternions, however, the monic and non-monic octonionic polynomials do not have always the same set of zeros.[4]
References
[edit]- ^ Eilenberg, Samuel; Niven, Ivan (April 1944). "The "fundamental theorem of algebra" for quaternions". Bulletin of the American Mathematical Society. 50 (4): 246–248. doi:10.1090/S0002-9904-1944-08125-1.
- ^ Liu, Ming-Sheng; Xiang, Na; Yang, Yan (2017). "On the Zeroes of Clifford Algebra-Valued Polynomials with Paravector Coefficients". Advances in Applied Clifford Algebras. 27 (2): 1531–1550. doi:10.1007/s00006-016-0748-9. ISSN 0188-7009. S2CID 253598676.
- ^ Jou, Yuh-Lin (1950). "The "fundamental theorem of algebra" for Cayley numbers". Acad. Sinica Science Record. 3: 29–33.
- ^ Serôdio, Rogério (2007). "On Octonionic Polynomials". Advances in Applied Clifford Algebras. 17 (2): 245–258. doi:10.1007/s00006-007-0026-y. ISSN 0188-7009. S2CID 123578310.
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