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Results tagged with polynomials
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user 50509
Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.
3
votes
Transforming numbers of irreducible polynomials
Here is a reformulation of Ofir Gorodetsky's excellent answer to my question:
Let $(a(n))_{n \geq 1}$ be a sequence of rational numbers. Define
the transformed sequence $T(a)$ of $a$ to have $n$th ele …
- 481
4
votes
2
answers
612
views
Transforming numbers of irreducible polynomials
Let $M(n)(q)$, where $q$ is a prime power and $n$ a natural number,
stand for the number of irreducible monic polynomials of degree $n$ in the
polynomial ring $\mathbf{F}_{q}[X]$ over the finite field … Consider the sum of polynomials in
$\mathbf{Q}[q]$
\begin{equation*}
\sum_{[n_1^{e_1},\ldots, n_s^{e_s}] \in P(n)} (-1)^{\sum e_i}
\prod_{i=1}^s \binom{M(n_i)}{e_i}
\end{equation*}
ranging over all …
- 481
8
votes
3
answers
376
views
Self-reciprocal polynomials over finite fields
Let $SRMI_q(2n)$ denote the number of self-reciprocal
irreducible monic polynomials of even degree $2n$ over the finite
field $\mathbf{F}_q$ with $q$ elements. … have in mind is an analogue of the well-known formula
\begin{equation*}
\prod_{n \geq 1} \frac{1}{(1-x^n)^{I_q(n)}} = \frac{1}{1-qx}
\end{equation*}
where $I_q(n)$ is the number of irreducible monic polynomials …
- 481