Anzeige
Erschienen in:
2008 | OriginalPaper | Buchkapitel
A Criterion for Polynomials to Divide Infinitely Many k- Nomials
verfasst von : Lajos Hajdu, Robert Tijdeman
Erschienen in: Diophantine Approximation
Verlag: Springer Vienna
Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
A polynomial
Q ∈
ℚ[
x
] of the form
$$ Q\left( x \right) = \sum\limits_{i = 1}^k {a_i x^{m_i } } with m_1 > \ldots > m_{k - 1} > m_k = 0 and a_1 = 1 $$
is called a standard
k
-nomial. It is worth to mention that the restriction to monic
k
-nomials is only for convenience. We may replace every standard
k
-nomial by any of its constant multiples, and the theorems would still be valid. We call
(m
1
, ...,
m
k
) the exponent
k
-tuple of
Q.
Note that if
Q
is a standard
k
-nomial, but not a standard (
k
-1)-nomial, then its exponent
k
-tuple is uniquely determined. Let
$$ \begin{gathered} PR_k = \left\{ {P \in \mathbb{Q}\left[ x \right]: \exists Q \in \mathbb{Q}\left[ x \right] and r \in \mathbb{Z} with deg \left( Q \right) < k} \right. \hfill \\ \left. {and r \geqslant 1 such that P\left( x \right)\left| { Q\left( {x^r } \right)over \mathbb{Q}} \right.} \right\}. \hfill \\ \end{gathered} $$