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Results tagged with polynomials
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user 2480
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.
10
votes
1
answer
1k
views
Is every positive polynomial the ratio of 2 positive coefficient polynomials?
I can prove that a quadratic positive polynomial is the ratio of 2 polynomials with non negative coefficients, for example $\displaystyle x^2-x+1/3=\frac{x^6+1/27}{x^4+x^3+2/3 x^2+1/3 x+1/9}$, and similarly … The full proof is not hard and involves some recursive polynomials related to Chebyshev polynomials of the second kind. …
- 12.9k
4
votes
Why do we make such big deal about the 'unsolvability' of the quintic?
Then naturally mathematicians became curious about whether the resulting enlarged family of numbers (or "field" in modern language) would
A) consist always of solutions of polynomials;
B) contain all such …
- 5,103
34
votes
1
answer
1k
views
Does any cubic polynomial become reducible through composition with some quadratic?
irreducible cubic polynomial $P(X)\in \mathbb{Z}[X]$ is there always a quadratic $Q(X)\in \mathbb{Z}[X]$ such that $P(Q)$ is reducible (as a polynomial, and then necessarily the product of 2 irreducible cubic polynomials …
- 1,042
9
votes
1
answer
990
views
Are polynomials bounded on the primes possible?
If $\{p_i\}$ is the sequence of all primes, is it possible that there exist a non constant $P\in \mathbb{Z}[x_1,\dots x_n]$ such that $P(p_i,p_{i+1},\dots p_{i+n-1})$ is bounded in $i$?
More precisely …
- 32.1k
1
vote
polynomials with similar maxima-minima
I think that some computing around shows that $(x^6+y^6)^5$ and $(x^{10}+y^{10})^3$, or $x^{12}+y^{12}$ and $(x^4+y^4)^3$ for that matter, have the same set of extremes. So there can't be an algebraic …
- 5,103
3
votes
q-th powers and roots of polynomials
A family of counterexamples is defined as follows:
$r=2$,
$p_1\ge 2, \quad p_2\ge 3, \quad d_1 \perp p_1, \quad d_2 \perp p_2$,
$a=\dfrac{\tan(\pi d_1/p_1)-\tan(\pi d_2/p_2)}{\tan(\pi d_1/p_1)+\t …
- 5,103
1
vote
Sum of two squares and implication of Bunyakovsky conjecture
Notice that in your example
$(a1-\textit{i}*a2)*(-4+3\textit{i})/5 = a3-\textit{i}*a4$.
$\mathbb{Z}[\textit{i}]$ is UFD and so is $\mathbb{Z}[\textit{i}][x]$.
- 5,103