# Tagged Questions

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-...

262 views

144 views

### Connection between the Chebyshev polynomials and the Faber polynomials

From a comment on this question: @draks, there is a connection between the Chebyshev polynomials and the Faber polynomials (a.k.a. Shur polynomials), which 'invert" the cyclic partition ...
56 views

### How to approximate higher-degree multivariate polynomial in space of lower-degree multivariate polynomials with some constraints?

For a polynomial $P_{1}(x)$, $x\in {\mathbb R}^n$ with a higher-degree, how to find a lower-degree polynomial $P_{2}(x)$ with determined structure or bounded degree to approximate it with the ...
324 views

### Is there a way to find out how many distinct roots a polynomial has? [closed]

Let say we have an arbitrary polynomial over the reals, and we do not know whether it is separable or not. Is there some algorithm to find the number of roots it has in the complex number?
442 views

411 views

Let $c(n)$ in $\mathbb{Z}/2\mathbb{Z}[x]$ be defined by the recursion $$c(n+4)=c(n+3)+(x^4+x^3+x^2+x)c(n)+x^n\cdot(x+x^2),$$ and the initial conditions $$c(0)=0,\quad c(1)=1,\quad c(2)=x,\quad c(3)=x^... 0answers 114 views ### Order of zeros for sparse polynomials mod p It is a fairly well known fact that sparse polynomials f(x) cannot have large order zeros other than at x=0. If f(x)=a_1x^{r_1}+\cdots+a_kx^{r_k} then at c \neq 0, f has a zero of order at ... 0answers 217 views ### Endofunctors on the category of groups which are Galois- related to a linear map on \mathbb{Q}[x] I thank J.C. and Arturo Magidin for interesting suggestions in the comments to this question In this post the field of rational numbers is denoted by \mathbb{Q}. The space of polynomials with ... 0answers 307 views ### Aligned roots of irreducible polynomials It is well known from this famous question that the roots of a random polynomial tend to be close to the unit circle. So I was wondering in a somewhat converse sense: for an irreducible polynomial, is ... 1answer 200 views ### Do semialgebraic sets depend outer semicontinuously on their defining polynomials? Consider a compact (semialgebraic) ball B\subset \mathbb{R}^n and a semialgebraic set A=A(f_1,\cdots,f_s)\subset \mathbb{R}^n defined through some representation in terms of polynomials f_1,\... 0answers 128 views ### On factorization algorithms for \mathcal{O}[x] We know that \mathsf{LLL} algorithm provides factorization procedure that runs in poly time for polynomials in \Bbb Z[x] that are primitive. What other rings \mathcal{O} can we use instead of \... 2answers 547 views ### Is there a \mathbb{Q}-linear map T over \mathbb{Q}[x] such that for all polynomials Gal(T(f)) \simeq The commutator subgroup of Gal(f)? I asked this question at MSE but I did not receive an answer. So I ask it at MO: We denote the field of rational numbers by \mathbb{Q}. The Galois group of a polynomial f is denoted by Gal(f)... 1answer 375 views ### Injectivity of a multivariate homogeneous polynomial mapping Consider the mapping$$ \Psi: \mathbb R^2 \to \mathbb R^5, \\ \Psi(x) = \begin{pmatrix} x_1 \\ x_2 \\ x_1^2 \\ x_1 x_2 \\ x_2^2 \end{pmatrix}.$$Which are the matrices A \in \mathbb R^{m \times 5} ... 2answers 409 views ### Why are most coefficients of these minimal polynomials divisible by p? For an odd prime p, let \zeta:=e^{\frac{\pi i}p} and choose odd 1<n<p. Further let q(x) and r(x) be integer polynomials such that r(x) has no common factor with x^n+1, and \xi ... 1answer 171 views ### Determinant of symmetric Latin square Let n=2m be an even number. Let us construct n\times n symmetric matrices S_n in the following way. The entries are indeterminates X_1,\ldots,X_{n-1}. We choose a 1-factorization of the ... 1answer 500 views ### An elementary lower bound on the number of primes Recall the second Chebyshev function:$$\psi(x) = \sum_{p \leq x} \lfloor \log_p x \rfloor \log p where $x$ is a positive integer, and $p$ runs over all primes $\leq x$. In a hunt for an "...
For the positive integer $m$, let $n=\varphi(m)$ denote the totient of $m$. Given a rational prime $p$ such that $p\equiv 1 \bmod m$, let $A=\{a_1,\cdots, a_n\}\subseteq \mathbb{F}_p$ denote the set ...