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Results tagged with polynomials
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user 6153
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.
8
votes
What Are Some Naturally-Occurring High-Degree Polynomials?
Cyclotomic polynomials (http://en.wikipedia.org/wiki/Cyclotomic_polynomial), which when n is composite are rather more complicated than you might guess. … It is possible to reason about polynomials without knowing exactly what the solutions look like. …
- 12.6k
7
votes
What, if anything, makes homogeneous polynomials so great?
Well, if you are interested in counting solutions mod p, you could note that the "good" formulae are indeed related to the homogeneous approach/projective space. It is not just a question of restoring …
- 12.6k
5
votes
Which numbers appear as discriminants of cubics?
The discriminants of cubic polynomials are there explained in terms of discriminants of cubic fields, and asymptotics given for the latter. …
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2
votes
Generators of cyclic group of finite fields
See for example https://en.wikipedia.org/wiki/Conway_polynomial_(finite_fields). There is plenty of theory and practice in the area, going back to Richard Parker.
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1
vote
Is $x^p-x+1$ always irreducible in $\mathbb F_p[x]$?
In the notation of https://en.wikipedia.org/wiki/Berlekamp%27s_algorithm, the point is that the space of polynomials g congruent to their p-th power is not going to contain anything but constants, because …
- 12.6k
1
vote
Roots of a polynomial in several variables
You need as many polynomials as variables to have a finite number. (There is much, much more to say.) …
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1
vote
A question regarding polynomials whose roots satisfy certain algebraic relation
Such questions formed a substantial part of the classical "theory of equations", before Galois theory was formulated. For example, there is a book by Burnside on theory of equations, that is easy to f …
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1
vote
minimal polynomials of trig functions of ($k \pi/p$) and divisibility of coefficients by p
I can't quite do this all in my head, but clearly enough you should should write 1 - ζ = π and do π-adic analysis in the cyclotomic field of p-th roots of unity, where ζ is a non-trivial p-th root of …
- 12.6k
0
votes
Accepted
Probability of summing products of irreducible polynomials in a finite field to zero
Taking a view from old-fashioned algebra: your sum can be interpreted from a generating function of a product of three geometric series. That generating function has a partial fraction decomposition, …
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