# Questions tagged [polynomials]

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.

1,534 questions
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### Formula for the index of regularity of a generic Hilbert function

Is there an explicit formula for the index of regularity of a generic Hilbert function in two variables? (i.e., the Hilbert function of an ideal of $k[X,Y]$ generated by $r$ generic forms $f_{i}$ of ...
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### Infinitude of cyclotomic polynomials with a certain number of terms

Let $\Phi_n$ be the $n$th cyclotomic polynomial: $${\Phi _{n}(x)=\!\!\prod _{\substack {1\leq k\leq n \\ \gcd(k,n)=1}} \!\!\big(x-e^{2i\pi {k/n}}\big).}$$ Here is a list of the first 30 cyclotomic ...
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### Trees and Shabat polynomials

Recently, I read the relation between Shabat polynomials and trees. The book [0] says that if $p: \mathbb{C} \to \mathbb{C}$ is a degree-$n$ polynomial such that the segment $[-1,1]$ contains no ...
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### Kernel of evaluation map into field of quotients

Let $R$ be an integral domain and for $a \in R$ denote by $\text{eval}_a: R[X] \to R$ evaluation at $a$. It's well-known (and easy to see) that $$\ker(\text{eval}_a)=(X-a).$$ The next more ...
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### Question regarding the image of a polynomial map containing a small box

I have the following question, which intuitively seems it should be true but I wasn't sure how to prove it rigorously. Let $\delta, \varepsilon > 0$. Let $\Psi_i(w_1, w_2, \mathbf{v})$ be a ...
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Are there infinitely many integers $n>0$ at which a $0$ and $\pm1$ coefficient polynomial $f(x)$ that divides $x^{m}-1$ (thus $f(x)$ is product of cyclotomials) with degree $m-O(n^{})$ where $m=\... 1answer 124 views ### Is there any pseudoprime that pass this test above tested range, or any prime that does not show these ending patterns? if the recurrence sequence is defined by the following foormula,$d_{n + 3} = 3d_{n + 2} - d_{n + 1} - 2d_n$where$d_1 = 1, d_2 = 3$and$ d_3 = 7$, this produce the following complex sequence $$1, ... 1answer 122 views ### Semialgebraic sets containing irrational power functions Let \alpha be an irrational number, and consider the set A=\{(x,x^\alpha),x\ge 0\}\subseteq \mathbb{R}^2, which is the graph of the function f(x)=x^\alpha. I'm trying to prove/disprove the ... 1answer 109 views ### Locus of roots of all convex combinations of two monic polynomials, II This post contains a revised conjecture to a conjecture I posed previously which was shown to be false. Let p, q \in \mathbb{C}[t] be two monic polynomials of degree n \ge 1. For \alpha \in [0,1]... 2answers 373 views ### is there any non prime (pseudoprime) that holds true for this test, or any prime that fall out of this test? The sequence is defined by the following formula d_{n + 3} = d_{n + 2} + 2d_{n + 1} - d_n where d_1 = 0, d_2 = 1, d_3 = 2, \{0, 1, 2, 4, 7, 13, 23,42, ...\} if this sequence is calculated over ... 0answers 32 views ### Root separation of polynomials with the same coefficients Consider two polynomials of the following form:$$f(x)=k_1 x^{a_1}+\ldots+k_d x^{a_d}g(x)=k_1 x^{b_1}+\ldots+k_d x^{b_d}$$with the following properties: k_1>0 For every i, if k_i>0 ... 1answer 163 views ### Second order recurrence relation for third order polynomial root Consider this recurrence relation:$$ \begin{eqnarray*} f_0&=&1\\ f_n&=& \sum_{m=0}^{n-1} \frac{\left(\frac{m+3}{2}\right)_{m-1}}{\left(\frac{m+2}{2}\right)_m} f_{n-m-1} f_m\ \ \ \... 2answers 665 views ### Zeros of MacLaurin polynomials for the exponential function Asked but never answered at MSE. Let$\exp_n(z)$denote the nth degree Taylor polynomial of$e^z$:$\exp_n(z) = 1 + z + z^2/2! + ... + z^n/n! \;$. The zeros of$\exp_n(z)$were studied by ... 0answers 162 views ### What is known about the coefficients of high powers of the Vandermonde determinant? This question is about high powers (parameter$p$going to infinity) of the Vandermonde determinant with$N$entries, for fixed$N$. I expand $$\Delta(X_1, \dots, X_N)^p = \prod_{i < j} (X_i - X_j)... 0answers 76 views ### Does there exist a triplet of polynomials with natural coefficients with the following properties? Does there exist a triplet (f,g,h) of polynomials with non-negative integer coefficients a_i,b_i,c_i, for chosen degree p\gt1$$f(x)=a_{p}x^{p}+a_{p-1}x^{p-1}+\dots+a_{1}x^{1}+a_{0}g(x)=b_{... 1answer 157 views ### How does the minimal degree of a monic polynomial with all values divisible by$p^n$asymptotically behave? Let$p$be a prime number. For every$n \in \mathbb N$, let$A_{p,n}:=\{\deg P(X) : P(X)\in \mathbb Z[X]$is monic and$p^n|P(m), \forall m \in \mathbb Z\}$. As user abx notes below,$A_{p,n}$... 1answer 426 views ### Minimal cardinality of a field where a polynomial has a root Let$P(n)$be the set of all monic polynomials of degree$n$with integer coefficients, such that all coefficients have absolute value at most$2^n$. Given a positive integer$n$let us define$A(n)$... 1answer 468 views ### Cyclotomic polynomials 2 My naive question may actually lead to something interesting. Let$\Phi_m(x)$,$\Phi_n(x)$be cyclotomic polynomials,$m<n$. These polynomials are relatively prime and so there are polynomials$...
Let $\Phi_m(x)$ and $\Phi_n(x)$ be two different cyclotomic polynomials. Then $\Phi_m(x)$ and $\Phi_n(x)$ are coprime, so there are two polynomials $s(x), t(x)$ with, say, rational coefficients such ...
We know cyclotomic polynomials $\Phi_{2^kp^rq^m}(x)$ have coefficients in $\{0,\pm1\}$. What is the largest degree $f_{d,n}(x)=\frac{x^{2^{k}p^{r}q^{m}}-1}{\Phi_d(x)}$ with $\{0,\pm1\}$ coefficients ...