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Results tagged with polynomials
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user 2389
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.
1
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0
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261
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One-sided version of the "best approximation polynomial" : Upper polynomial approximations
Let $X$ be a finite subset of $\mathbb R$ and let $f : X \to {\mathbb R}$. Suppose we want to approximate $f$ by a polynomial $g$ of fixed degree $d\geq 1$ with the additional condition
$g\geq f$. Let …
- 3,595
3
votes
3
answers
595
views
Every positive polynomial with rational coefficients is above a completely Q-factorized nonn...
This question was originally asked in stackoverflow (https://math.stackexchange.com/questions/103941/every-positive-polynomial-is-above-a-completely-q-factorized-positive-polynomial) but as it has rem …
- 3,595
17
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4
answers
988
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Finite interpolation by a nondecreasing polynomial
It is well known that there are polynomials that "interpolate"
in that $f(x_i)=y_i$ for all $i$, and the Lagrange interpolating polynomial
even warrants a solution of degree $ < n$. …
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4
votes
2
answers
874
views
Is it true that all the "irrational power" functions are almost polynomial ?
Hello all, the $\Delta$ operator on functions $\mathcal{N} \to \mathbb{R}$
(where $\mathcal N$ denotes $\lbrace 1,2, \ldots , \rbrace$ )defined by
$\Delta(f)(n)=f(n+1)-f(n)$ is well-known and
it is …
- 3,595
8
votes
0
answers
492
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"Consecutive" irreducible polynomials
If $P\in {\mathbb Z}[X]$ is a polynomial of degree $2$, then
it is easy to see that for any integer $m$, at least one of the polynomials
$P-(m+1),P-(m+2),P-(m+3),P-(m+4)$ is irreducible in ${\mathbb Z} …
- 3,595
3
votes
3
answers
292
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Expressing field inclusions by polynomial equalities on coefficients
Let $A$ be the set of all quadruples $(a_0,a_1,a_2,a_3) \in {\mathbb Q}^4$ such that
the polynomial $P=X^4+a_3X^3+a_2X^2+a_1X+a_0$ is irreducible and if $z$ is any root
of $P$, then ${\mathbb Q}(z)$ c …
- 3,595
2
votes
2
answers
1k
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algebraic numbers of degree 3 and 6, whose sum has degree 12
This question is related to Degree of sum of algebraic numbers. Forgive me if
this is a dumb question, but are there two algebraic numbers $a$ and $b$ of degree $3$ and $6$ respectively, such that the …
- 3,595
5
votes
0
answers
190
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"Unknot" algebraic set defined by two mutually dependent set of variables
Let $n$ be an integer $\geq 4$, and let $V \subseteq {\mathbb C}^{2n-1}$ be the set of all
$(a_1,a_2, \ldots ,a_n,b_1,b_2, \ldots ,b_{n-1}) \in {\mathbb C}^{2n-1}$ such that the derivative of the pol …
- 3,595
12
votes
3
answers
2k
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Growth of the "cube of square root" function
Hello all, this question is a variant (and probably a more difficult one)
of a (promptly answered ) question that I asked here, at Is it true that all the "irrational power" functions are almost polyn …
- 3,595
6
votes
2
answers
604
views
What is the set of possible values of the degree of the sum of two algebraic numbers with fi...
This question is related to Degree of sum of algebraic numbers and algebraic numbers of degree 3 and 6, whose sum has degree 12.
In this last question I asked a very special case of the following p …
- 3,595
8
votes
1
answer
599
views
Integer polynomial (of degree >1) all of whose values are square-free
Is there an integer polynomial $ A \in {\mathbb Z} [ X ]$ of degree $d\geq 2$ such that for any integer $n\in {\mathbb Z}$ , $ A(n) $ is a square-free integer?
- 3,595
1
vote
2
answers
339
views
Infinite collection of elements of a number field with very similar annihilating polynomials
In other words, we are asking for infinitely many elements in $B$,
whose minimal polynomials are ``as similar as possible". …
- 3,595
15
votes
2
answers
566
views
Which polynomials arise as formulas for a conjugate
For any integer $r \geq 2$, et $V_r$ be the set of polynomials $Q \in {\mathbb Q}[X]$ of degree $r-1$ such that there is an algebraic number $\alpha$ of degree $r$ , such that
$Q(\alpha)$ is a conjugate … It is not hard to see that $V_2$ consists exactly of $X$ and all the polynomials $a_0-X$, for $a_0\in {\mathbb Q}$. Have
the $V_r(r \geq 3)$ been studied? Is anything known about $V_3$ ? …
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