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Results tagged with polynomials
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user 290
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.
14
votes
Accepted
Todd polynomials
.$$
As a sanity check, expanding the terms in WolframAlpha up to $t^4$ in terms of elementary symmetric polynomials / Chern classes gives
$$T_1 = \frac{c_1}{2}$$
$$T_2 = \frac{c_1^2 + c_2}{12}$$
$$T_3 …
- 118k
30
votes
1
answer
2k
views
Which of the proofs of the fundamental theorem of algebra can actually produce bounds on whe...
One of the old classic MO questions is a big-list of proofs of the fundamental theorem of algebra. Here is a second big-list question about this big list:
Which of the FTA proofs can, even in prin …
- 118k
3
votes
Link between Irreducible Factors and Prime Factors (or Cycles of a Permutation)
I wrote a blog post on this here. The basic result is that for fixed $k$ and $n$, as $q \to \infty$ the joint distribution of irreducible factors of degrees $1$ through $k$ of a random monic polynomia …
- 118k
22
votes
Accepted
Integer valued polynomial through some points with rational coordinates
The integer-valued polynomials have a basis (over $\mathbb{Z}$) given by the Newton polynomials
$$\displaystyle {x \choose n} = \frac{x (x - 1)\dots(x - (n-1))}{n!} …
- 118k
23
votes
Accepted
Is there an integer a such that f(X)+a is irreducible in Z[X]?
Yes, and you don't need $f$ irreducible. The following irreducibility criterion suffices and shows that infinitely many $a$ work.
Lemma: Let $g(x) = a_n x^n + ... + a_0 \in \mathbb{Z}[x]$ be such th …
- 118k
6
votes
Which positive definite symmetric matrices have solvable characteristic polynomial?
As a subspace of the space $\mathbb{Q}^n$ of monic polynomials of degree $n$ with rational coefficients, the solvable polynomials are dense (and so in particular are not contained in an algebraic or even … To see this it suffices to observe that any such polynomial is a product of real linear or quadratic polynomials and that we can approximate these by rational linear or quadratic polynomials. …
- 118k
50
votes
Accepted
Given a polynomial f, can there be more than one constant c such that every root of f(x)-c i...
This is impossible by the Mason-Stothers theorem (which holds over any algebraically closed field of characteristic zero).
We want to find $f, g, h$ such that $f + g = h$ where $g$ is a constant and …
- 118k
5
votes
Are plethories a theory of basis-free polynomials?
Some naive remarks. It seems to me that the simplest reason to choose the standard basis is because it exhibits the universal property of a polynomial ring.
One way to exhibit a partially basis-free …
- 118k
8
votes
If a polynomial f is irreducible then (f) is radical, without unique factorization?
No, in the sense that this statement is false in a ring without unique factorization. For example, the element $2 + \sqrt{-5}$ is irreducible in $\mathbb{Z}[\sqrt{-5}]$, and $9 \in (2 + \sqrt{-5})$ bu …
- 118k
7
votes
Can this nested sum be expressed in terms of generalized harmonic numbers and the cycle inde...
I'll edit in the details later, but this is essentially a consequence of Polya's enumeration theorem together with an inclusion-exclusion argument; see this blog post.
Edit: Okay, so it's a little e …
- 118k
2
votes
Examples of nice families of irreducible polynomials over Z
But I am not really sure what you want, since already Eisenstein's criterion lets you write down large parameterized families of irreducible polynomials. Can you be more specific? …
- 118k
13
votes
Accepted
Integer polynomial (of degree >1) all of whose values are square-free
No. WLOG $A$ is irreducible. Pick a sufficiently large prime $p$ dividing $A(k)$ for some $k$ (there are infinitely many such primes, for example by the argument here). In particular pick $p$ large …
- 118k
15
votes
Surprising behaviour of polynomial that generates the series 1,2,4,8,...2^(k-1)
The second observation is true of all polynomials which interpolate an integer sequence. …
- 118k
9
votes
Factorizing polynomials of several variables (in a different perespective)
As David indicates, the truth is much more interesting: the zero set of polynomials in two or more variables generically describe interesting geometric structures called algebraic varieties, and the case … in which the polynomials factor into linear factors corresponds to the most boring structure possible: a bunch of straight lines. …
- 118k
3
votes
Periodic orbits and polynomials
The irreducible polynomials of degree $n$ over $\mathbb{F}_2[x]$ can be identified with the periodic orbits of minimal period $n$ of the Frobenius map $x \mapsto x^2$ acting on $\overline{ \mathbb{F}_2 … It might be easier to give an explicit bijection to Lyndon words; for one thing, Lyndon words on alphabets of any size make sense whereas irreducible polynomials only make sense over prime power fields …
- 118k