The Prouhet–Tarry–Escott problem, indecomposability of polynomials and Diophantine equations
In this paper, we show how the subjects mentioned in the title are related. First we study the structure of partitions of A ⊆ { 1 , ⋯ , n } in k -sets such that the first k - 1 symmetric polynomials of the elements of the k -sets coincide. Then we apply this result to derive a decomposability result...
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| Vydáno v: | The Ramanujan journal Ročník 58; číslo 4; s. 1075 - 1093 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.08.2022
Springer Nature B.V |
| Témata: | |
| ISSN: | 1382-4090, 1572-9303 |
| On-line přístup: | Získat plný text |
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| Abstract | In this paper, we show how the subjects mentioned in the title are related. First we study the structure of partitions of
A
⊆
{
1
,
⋯
,
n
}
in
k
-sets such that the first
k
-
1
symmetric polynomials of the elements of the
k
-sets coincide. Then we apply this result to derive a decomposability result for the polynomial
f
A
(
x
)
:
=
∏
x
∈
A
(
x
-
a
)
. Finally we prove two theorems on the structure of the solutions (
x
,
y
) of the Diophantine equation
f
A
(
x
)
=
P
(
y
)
where
P
(
y
)
∈
Q
[
y
]
and on shifted power values of
f
A
(
x
)
. |
|---|---|
| AbstractList | In this paper, we show how the subjects mentioned in the title are related. First we study the structure of partitions of
A
⊆
{
1
,
⋯
,
n
}
in
k
-sets such that the first
k
-
1
symmetric polynomials of the elements of the
k
-sets coincide. Then we apply this result to derive a decomposability result for the polynomial
f
A
(
x
)
:
=
∏
x
∈
A
(
x
-
a
)
. Finally we prove two theorems on the structure of the solutions (
x
,
y
) of the Diophantine equation
f
A
(
x
)
=
P
(
y
)
where
P
(
y
)
∈
Q
[
y
]
and on shifted power values of
f
A
(
x
)
. In this paper, we show how the subjects mentioned in the title are related. First we study the structure of partitions of $$A \subseteq \{1, \dots , n\}$$ A ⊆ { 1 , ⋯ , n } in k -sets such that the first $$k-1$$ k - 1 symmetric polynomials of the elements of the k -sets coincide. Then we apply this result to derive a decomposability result for the polynomial $$f_A(x) := \prod _{x \in A} (x-a)$$ f A ( x ) : = ∏ x ∈ A ( x - a ) . Finally we prove two theorems on the structure of the solutions ( x , y ) of the Diophantine equation $$f_A(x)=P(y)$$ f A ( x ) = P ( y ) where $$P(y)\in \mathbb {Q}[y]$$ P ( y ) ∈ Q [ y ] and on shifted power values of $$f_A(x)$$ f A ( x ) . In this paper, we show how the subjects mentioned in the title are related. First we study the structure of partitions of A⊆{1,⋯,n} in k-sets such that the first k-1 symmetric polynomials of the elements of the k-sets coincide. Then we apply this result to derive a decomposability result for the polynomial fA(x):=∏x∈A(x-a). Finally we prove two theorems on the structure of the solutions (x, y) of the Diophantine equation fA(x)=P(y) where P(y)∈Q[y] and on shifted power values of fA(x). |
| Author | Tijdeman, R. Papp, Á. Hajdu, L. |
| Author_xml | – sequence: 1 givenname: L. surname: Hajdu fullname: Hajdu, L. organization: Institute of Mathematics, University of Debrecen, Alfréd Rényi Institute of Mathematics – sequence: 2 givenname: Á. orcidid: 0000-0002-4587-4428 surname: Papp fullname: Papp, Á. email: papp.agoston@science.unideb.hu organization: Institute of Mathematics, University of Debrecen – sequence: 3 givenname: R. surname: Tijdeman fullname: Tijdeman, R. organization: Mathematical Institute, Leiden University |
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| Cites_doi | 10.4064/aa-31-2-199-204 10.4007/annals.2020.191.2.2 10.1006/jnth.1997.2109 10.1016/S0019-3577(03)90069-3 10.4064/aa-95-3-261-288 10.4064/aa129-1-1 10.1023/A:1025480729778 10.1090/S0002-9947-1922-1501189-9 10.4064/aa113-4-1 10.1112/jlms/s1-26.3.176 10.4064/aa-80-3-289-295 10.1145/2644288.2644292 10.4064/aa99-4-5 10.1007/BF01974110 10.1016/j.jnt.2019.08.007 10.3390/math7030227 10.1093/qmath/12.1.304 10.1007/s00605-020-01422-7 10.1515/9783110285581.11 10.5486/PMD.2004.2913 |
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| Keywords | Products of consecutive integers Indecomposability of polynomials Symmetric polynomials The Prouhet–Tarry–Escott problem Polynomial values 11D41 11B75 Partitions of 11P05 |
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| PublicationSubtitle | An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan |
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| References | Saradha, Shorey (CR20) 2001; 99 Kulkarni, Sury (CR16) 2003; 14 CR15 CR13 Erdős (CR9) 1951; 26 Erdős, Selfridge (CR10) 1975; 19 Fried (CR11) 1973; 264 Raghavendran, Varayanan (CR18) 2019; 7 Brindza (CR6) 1984; 44 Fried (CR12) 1999 Blankertz (CR5) 2014; 48:1 Schinzel, Tijdeman (CR23) 1976; 31 de Weger (CR8) 1997; 63 CR2 Bilu, Tichy (CR4) 2000; 95 Ritt (CR19) 1922; 23 Saradha, Shorey (CR21) 2003; 138 CR7 Saradha, Shorey (CR22) 2007; 129 Lidl, Mullen, Turnwald (CR17) 1993 CR26 CR25 CR24 Bilu, Kulkarni, Sury (CR3) 2004; 113 Bennett, Siksek (CR1) 2020; 191 Győry (CR14) 1997; 80 JF Ritt (555_CR19) 1922; 23 555_CR24 555_CR2 555_CR7 R Blankertz (555_CR5) 2014; 48:1 555_CR26 555_CR25 P Erdős (555_CR10) 1975; 19 N Saradha (555_CR20) 2001; 99 N Saradha (555_CR21) 2003; 138 R Lidl (555_CR17) 1993 A Schinzel (555_CR23) 1976; 31 K Győry (555_CR14) 1997; 80 Yu Bilu (555_CR3) 2004; 113 S Raghavendran (555_CR18) 2019; 7 M Fried (555_CR12) 1999 555_CR13 N Saradha (555_CR22) 2007; 129 P Erdős (555_CR9) 1951; 26 B de Weger (555_CR8) 1997; 63 555_CR15 B Brindza (555_CR6) 1984; 44 M Bennett (555_CR1) 2020; 191 M Fried (555_CR11) 1973; 264 M Kulkarni (555_CR16) 2003; 14 Yu Bilu (555_CR4) 2000; 95 |
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Algebra doi: 10.1145/2644288.2644292 – ident: CR7 – year: 1993 ident: CR17 publication-title: Dickson Polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics – volume: 99 start-page: 363 year: 2001 end-page: 388 ident: CR20 article-title: Almost perfect powers in arithmetic progression publication-title: Acta Arith. doi: 10.4064/aa99-4-5 – ident: CR26 – volume: 44 start-page: 133 year: 1984 end-page: 139 ident: CR6 article-title: On -integral solutions of the equation publication-title: Acta Math. Hungar. doi: 10.1007/BF01974110 – ident: CR24 – volume-title: Dickson Polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics year: 1993 ident: 555_CR17 – volume: 44 start-page: 133 year: 1984 ident: 555_CR6 publication-title: Acta Math. Hungar. doi: 10.1007/BF01974110 – ident: 555_CR25 – ident: 555_CR13 doi: 10.1016/j.jnt.2019.08.007 – volume: 7 start-page: 227 year: 2019 ident: 555_CR18 publication-title: MDPI Math. doi: 10.3390/math7030227 – volume: 19 start-page: 292 year: 1975 ident: 555_CR10 publication-title: Ill. J. Math. – ident: 555_CR7 doi: 10.1093/qmath/12.1.304 – volume: 113 start-page: 303 year: 2004 ident: 555_CR3 publication-title: Acta Arith. doi: 10.4064/aa113-4-1 – volume: 26 start-page: 176 year: 1951 ident: 555_CR9 publication-title: J. Lond. Math. Soc. doi: 10.1112/jlms/s1-26.3.176 – volume: 80 start-page: 289 year: 1997 ident: 555_CR14 publication-title: Acta Arith. doi: 10.4064/aa-80-3-289-295 – volume: 95 start-page: 261 year: 2000 ident: 555_CR4 publication-title: Acta Arith. doi: 10.4064/aa-95-3-261-288 – ident: 555_CR26 – volume: 48:1 start-page: 13 issue: 187 year: 2014 ident: 555_CR5 publication-title: ACM Commun. Comput. Algebra doi: 10.1145/2644288.2644292 – volume: 264 start-page: 40 year: 1973 ident: 555_CR11 publication-title: J. Reine Angew. Math. – ident: 555_CR15 doi: 10.1007/s00605-020-01422-7 – volume: 31 start-page: 199 year: 1976 ident: 555_CR23 publication-title: Acta Arith. doi: 10.4064/aa-31-2-199-204 – volume: 63 start-page: 373 year: 1997 ident: 555_CR8 publication-title: J. Number Theory doi: 10.1006/jnth.1997.2109 – ident: 555_CR2 doi: 10.1515/9783110285581.11 – volume: 23 start-page: 51 year: 1922 ident: 555_CR19 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1922-1501189-9 – volume: 138 start-page: 113 year: 2003 ident: 555_CR21 publication-title: Compositio Math. doi: 10.1023/A:1025480729778 – ident: 555_CR24 doi: 10.5486/PMD.2004.2913 – volume: 99 start-page: 363 year: 2001 ident: 555_CR20 publication-title: Acta Arith. doi: 10.4064/aa99-4-5 – volume: 14 start-page: 35 year: 2003 ident: 555_CR16 publication-title: Indag. Math. doi: 10.1016/S0019-3577(03)90069-3 – volume: 129 start-page: 1 year: 2007 ident: 555_CR22 publication-title: Acta Arith. doi: 10.4064/aa129-1-1 – volume: 191 start-page: 355 year: 2020 ident: 555_CR1 publication-title: Ann. Math. doi: 10.4007/annals.2020.191.2.2 – start-page: 169 volume-title: Variables Separated Polynomials, the Genus 0 Problem and Moduli Spaces, Number Theory in Progress year: 1999 ident: 555_CR12 |
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| Snippet | In this paper, we show how the subjects mentioned in the title are related. First we study the structure of partitions of
A
⊆
{
1
,
⋯
,
n
}
in
k
-sets such... In this paper, we show how the subjects mentioned in the title are related. First we study the structure of partitions of $$A \subseteq \{1, \dots , n\}$$ A ⊆... In this paper, we show how the subjects mentioned in the title are related. First we study the structure of partitions of A⊆{1,⋯,n} in k-sets such that the... |
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| SubjectTerms | Combinatorics Diophantine equation Field Theory and Polynomials Fourier Analysis Functions of a Complex Variable Mathematics Mathematics and Statistics Number Theory Polynomials |
| Title | The Prouhet–Tarry–Escott problem, indecomposability of polynomials and Diophantine equations |
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