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
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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
$$ \begin{gathered} PR_k = \left\{ {P \in \mathbb{Q}\left[ x \right]: \exists Q \in \mathbb{Q}\left[ x \right] and r \in \mathbb{Z} with deg \left( Q \right) < k} \right. \hfill \\ \left. {and r \geqslant 1 such that P\left( x \right)\left| { Q\left( {x^r } \right)over \mathbb{Q}} \right.} \right\}. \hfill \\ \end{gathered} $$