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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| Veröffentlicht in: | The Ramanujan journal Jg. 58; H. 4; S. 1075 - 1093 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
New York
Springer US
01.08.2022
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 1382-4090, 1572-9303 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | 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
)
. |
|---|---|
| Bibliographie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1382-4090 1572-9303 |
| DOI: | 10.1007/s11139-022-00555-7 |