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.
2,555
questions
1
vote
0
answers
200
views
Is there a natural topology on $\mathbb{C}(t)[x_1,\ldots, x_n]$ with this property?
Is there a good topology on $A=\mathbb{C}(t)[x_1,\ldots, x_n]$ so that $A$ is a topological algebra with the following property:
For any $N>0$ and a polynomial $F\in\mathbb{C}[x_1,\ldots, x_n]$ ...
1
vote
1
answer
214
views
Closed-form formula for a multivariate polynomial
Counting certain walks in threshold graphs, I came to the following independent problem. Assume that $x_1,\dots,x_a$ are independent variables and for $k\geq 2$ let
$$
P_k(x_1,\dots,x_a)=\sum_{(i_1,\...
1
vote
0
answers
225
views
Polynomial resultants restricted to intervals
The resultant of two polynomials, $R(f,g)$, is a polynomial in the coefficients of $f$ and $g$, and has the property that $R(f,g) = 0$ if and only if $f$ and $g$ share a common root (possibly in an ...
0
votes
0
answers
115
views
Maximum number of integer solutions with some size constraints to bivariate polynomials?
Take a bivariate polynomial of total degree $d$ satisfying $d=d_x=d_y>1$ in $\mathbb Z[x,y]$ with coefficients bound in absolute value by $b$ ($d_x$ is $x$-degree and $d_y$ is $y$-degree).
Given a ...
1
vote
0
answers
87
views
Distribution of number of integer solutions in box to bivariate polynomials?
Take a bivariate polynomial of degree $d_x+d_y>\max(d_x,d_y)>1$ in $\mathbb Z[x,y]$ with coefficients bound in absolute value by $b$ ($d_x$ is $x$-degree and $d_y$ is $y$-degree).
What is the ...
1
vote
0
answers
80
views
Completion of $K$-algebra of finite type with respect to the residue norm
Let $K$ be a non-archimedean field. For $n \in \mathbb{N}$ let
\begin{equation*}
T_n=\{ \sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in K[\![X_1,\ldots, ...
2
votes
1
answer
541
views
A question on a Macaulay2 computation
I have an ideal $I$ generated by quadratic and cubic homogeneous polynomials in $10$ variables.
Macaulay2 tells me that $I$ defines an irreducible variety $X$ of dimension $5$ and degree $10$ in $\...
4
votes
1
answer
450
views
All rational periodic points
I am trying to find all rational periodic points of a polynomial. To specify: a periodic point is the point that satisfy $f^n(x)=x$. It is related to dynamical systems in fact. So the current codes ...
2
votes
3
answers
363
views
Minimal polynomial in $\mathbb Z[x]$ of seventh degree with given roots
I am looking for a seventh degree polynomial with integer coefficients, which has the following roots.
$$x_1=2\left(\cos\frac{2\pi}{43}+\cos\frac{12\pi}{43}+\cos\frac{14\pi}{43}\right),$$
$$x_2=2\left(...
26
votes
1
answer
1k
views
Is the derivative of $x^n + x^{n-1} + \dots + x + 1$ irreducible?
I am working on some combinatorics problems. One of my problems leads to the following question:
Is it true that the derivative of $x^n + x^{n-1} + \dots + x + 1,$ namely $nx^{n-1} + (n-1)x^{n-2} + \...
22
votes
3
answers
2k
views
Discriminant of characteristic polynomial as sum of squares
The characteristic polynomial of a real symmetric $n\times n$ matrix $H$ has $n$ real roots, counted with multiplicity.
Therefore the discriminant $D(H)$ of this polynomial is zero or positive.
It is ...
19
votes
0
answers
520
views
univariate integer version of Hilbert's 17th problem
Let $f(x)$ be a polynomial of degree $d$ with integer coefficients such that $f(x)\geqslant 0$ for all real $x$. Is it necessarily true that there exists an integer $N(d)$ such that $N(d)\cdot f$ is a ...
6
votes
0
answers
323
views
Galois groups associated to matrices
When $A\in M_n(\mathbb{Q})$, we consider the pencil $A-xA^T$. Then $p_A(x)=\det(A-xA^T)$ is a self-reciprocal polynomial. $p_A$ can only be irreducible if $n=2p$ is even.
Question: For every $p$, does ...
0
votes
1
answer
40
views
Upper bound of a uniformly converging sequence of polynomials
Let $k\geq 2$, and let $P_k$ be a sequence of polynomials, such that:
$P_k=\sum_{n=2}^{k+1}a_{n,k}X^n \in \mathbb{Q}[X]$, $a_{2,k}\neq 0$, $\deg P_k \leq k+1$, and consider $P_k :[0,1]\rightarrow \...
9
votes
1
answer
987
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,...
1
vote
1
answer
318
views
Minimum number of generators of the product of ideals
Let $k$ be an algebraically closed field, let $I, J \subseteq k[x_1,\dots, x_n, y_1,\dots, y_m]$ be ideals, and let $i,j$ be the minimum number of generators of $I$ and $J$, respectively. I have two ...
4
votes
0
answers
129
views
$\delta$-equidistributed polynomials over finite fields
I'm trying to show that a polynomial over finite (prime) field is "close enough" to being equidistributed over its range. A polynomial $p(\cdot)$ from $\mathbb{F}^n$ to $\mathbb{F}$ is $\...
4
votes
1
answer
617
views
Jacobian criterion for algebraic independence over a perfect field in positive characteristics
It is well known that the Jacobian criterion for algebraic independence does not hold in general for fields of positive characteristics. However, the following partial statement seems promising:
...
6
votes
3
answers
490
views
Uniformly approximating a function and its derivative using polynomials
I'm struggling either proving or disproving the following statement:
Let $K\subset \mathbb{R}$ be compact, and $S = \mathrm{span}\{p_k, k = 0, 1, \ldots\}$, where $p_k$'s are polynomials over $K$. If ...
3
votes
0
answers
76
views
Is $X$ closed in $Aut_{\mathbb{C}(t)}(\mathbb{C}(t)[x_1,\ldots, x_n])$?
Consider $\mathbb{C}$-algebras
$$A = \mathbb{C}[t][x_1,\ldots, x_n]\subset\mathbb{C}(t)[x_1,\ldots, x_n] = B$$
Group $\operatorname{Aut}_{k(t)}(k(t)[x_1,\ldots, x_n])$ carry a power series topology (...
3
votes
1
answer
901
views
Coppersmith bivariate polynomial roots implementation
Given $f(x,y)\in\mathbb Z[x,y]$ Coppersmith in https://link.springer.com/chapter/10.1007%2F3-540-68339-9_16 provides a provable method to find integer roots in polynomial time and this method was also ...
10
votes
1
answer
994
views
SOS polynomials with rational coefficients
Suppose we are given a univariate polynomial with rational coefficients, $p \in \Bbb Q [x]$, and are told that $p$ can be expressed as the sum of $k$ squares of polynomials with rational coefficients. ...
0
votes
0
answers
44
views
On infinitely many unimodular integers of certain constraints
Given $r>0$ are there integers $w,x,y,z>r$ with $wz-xy=1$ and $2(x^2+z^2)=(w^2-y^2)$ and $(x^2-z^2)\not\equiv0\bmod(w^2-y^2)$?
How small can they be?
2
votes
2
answers
151
views
Is it possible to create a polynomial $p(x)$ with this relation between $p(0)$ and $p(c)$? -- Part 2
Note: This question is based on a previous question
I was continuing my research from last time, and I realized my question was too strict! Instead of the polynomial being strictly increasing, it only ...
8
votes
1
answer
247
views
$\operatorname{SL}_2(k)$ invariant polynomials in $k[x_1,x_2,y_1,y_2]$
Let $k$ be a field and let $\operatorname{SL}_2(k)$ act on $k[x_1,x_2]$ and $k[y_1,y_2]$ in the usual ways. These actions induce an action on the tensor product $k[x_1,x_2,y_1,y_2]$ that preserves ...
9
votes
0
answers
673
views
Van der Pol's identity for the sum of divisors and a quartic polynomial equation for odd perfect numbers
In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following recurrence relation ($n>1$):
$$n^2(n-1) = \frac{6}{\sigma(n)} \sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\...
0
votes
0
answers
60
views
Does base point free linear system of polynomials generate higher degree polynomials
Let $k$ be an algebraically closed field. Let $S=k[x_0,\ldots, x_4]$ be the ring of polynomials. We set $S^i$ to the graded piece of degree $i$ polynomials. Let $H$ be a hyperplane of $S^5$ with no ...
2
votes
1
answer
215
views
Using Nelder-Mead to solve system of polynomial equations
I am trying to solve a system of $9$ polynomial equations in $9$ unknowns over the non-negative reals.
Since the equations are quite large and I would like to use VBA, I prefer an algorithm that ...
3
votes
1
answer
183
views
Solutions to nonhomogeneous quadratic equation mod $N$
Is there any way to find non-trivial solutions to the equation $x^2 + y^2 - x \equiv 0 \mod{N}$? There are clearly several trivial solutions, for example $(x, y) = (0, 0), (1, 0), (2^{-1}, 2^{-1}), (2^...
0
votes
0
answers
53
views
Dependence of the complexity of solving polynomial sytems on the multidegree
Let $f_1,\ldots,f_n\in \mathbb{Q}[X_1,\ldots,X_n]$ be a system of $n$ polynomials in $n$ indeterminant, which only has finitely many solutions. Supose that the each of the variables $X_i$ appears at ...
7
votes
1
answer
389
views
Perfect numbers, Galois groups and a polynomial
Let $f(n,t) = \sum_{k=0}^{r-1} d_k t^k$ where $D_n = \{d_0=1,d_1,\cdots,d_{r-1}\}$ are all divisors of $n$.
For instance
$$f(28,t) = 28 t^{5} + 14 t^{4} + 7 t^{3} + 4 t^{2} + 2 t + 1$$
For even ...
0
votes
1
answer
93
views
$k[h(x),y] \subseteq k[h(x),y] + \langle h(x),y \rangle \subseteq k[x,y]$
Let $k$ be a field of characteristic zero.
Let $h=h(T) \in k[T]$ with $\deg(h)=d \geq 2$ and $h(0)=0$ (namely, $h$ has zero constant term).
Consider the following chain of $k$-algebras:
$$k \subseteq ...
8
votes
3
answers
598
views
Is it possible to create a polynomial $p(x)$ with this relation between $p(0)$ and $p(c)$?
Given $b$ and $c$ with $b,c>1$, is it possible to construct a polynomial $p(x)$, whose degree is $n$ for all $c$ and $b$, such that:
$|p|$ is strictly increasing on $[1,c]$
and $|b \cdot p(c)| &...
2
votes
0
answers
68
views
division of polynomials [closed]
Consider the set of polynomials $k[x^1,\cdots,x^n]$ over any field $k$. Now, given $p,q\in k[x^1,\cdots,x^n]$, what are the necessary and suficient conditions in order to q divide p? That is: $\exists ...
1
vote
0
answers
130
views
Cyclotomic polynomial written as $x^d g\left(x + \frac{1}{x}\right)$
It is known that for every palindromic polynomial $f(x)$ of even degree $2d$ there is a polynomial $g$ of degree $d$ such that
$f(x) = x^d g\left(x + \frac{1}{x}\right)$.
For $n>2$ cyclotomic ...
1
vote
0
answers
84
views
Can I expand the coefficients of these two (multivariate) polynomials?
I do have two multivariate polynomials that I want to set to equality to extract some parameters. For that, I want to equate the coefficients of these polynomials, but they are not given in standard ...
2
votes
0
answers
130
views
Question about Zariski cancelation problem
If $\mathbb{Q}[t_1,\ldots, t_n] = A[x_1,\ldots, x_{n-1}]$ has it been proven that $A\cong\mathbb{Q}[t]$?
6
votes
0
answers
323
views
Irreducibility of a palindromic polynomial
I have strong reasons to believe that the palindromic polynomial $p_n(x)$ defined by
$$p_n(x) = x^{2n}+2x^{2n-1}+3x^{2n-2}+ \cdots+ nx^{n+1}+(n+3)x^{n}+nx^{n-1}+\cdots+2x+1$$
is irreducible in $\...
3
votes
2
answers
195
views
Request for books/articles on random polynomials
Can somebody kindly recommend me a couple of introductory books/articles on random polynomials with clear expositions of fundamental results (like the distribution of roots, expected number of real ...
1
vote
1
answer
236
views
Combinatorics and geometry underlying a refined Pascal matrix/Newton identities
The partition polynomials of OEIS A263633 give the coefficients of the power series/o.g.f of the multiplicative inverse (reciprocal) of a power series/o.g.f. and so give the Newton identities for ...
2
votes
0
answers
68
views
Nonvanishing criterion for a polynomial of polynomials
Let $P \in \mathbb{F}_q[x_1, \dots, x_n][y]$ be some fixed polynomial. Substituting polynomials for the variables in $P$ gives a map $\mathbb{F}_q[x]^{n+1} \to \mathbb{F}_q[x]$ defined by
$$(Q_1(x), \...
0
votes
1
answer
126
views
Prove that the following running average is monotonically decreasing
Let $S_n$ be defined as $\frac{1}{n}\sum_{t=1}^{t=n} [x^2+(p-q)x]$ where $x = 1-(1-p-q)^t$. We want to find the conditions on $p$ and $q$ such that $S_n$ is monotonically decreasing for all $n$. $0 &...
-3
votes
1
answer
139
views
Doubt about lemma for polynomial equivalence [closed]
Multivariate polynomials $f,g$ are equivalent if there exists
invertible linear transformation $A$ such that $f(X)=g(A\cdot X)$
From paper p.1:
Lemma 1.1. (Structure of quadratic polynomials). Let $F$...
1
vote
0
answers
34
views
Mismatching degrees and # derivatives in polynomial interpolation error formula
It is well known that if $f : [a,b] \to \mathbb{R}$ is $n+1$ times differentiable and $p(x)$ denotes the polynomial interpolant to $f(x)$ in the $n+1$ points $\bigl(x_k \in [a,b]\bigr)_{k = 1}^{n+1}$, ...
7
votes
0
answers
198
views
"Universal" polynomial of bounded norm on the sphere
Consider the space $V_{d,n}=\mathbb{R}[x_1,\ldots,x_n]_d$ of homogeneous polynomials of degree $d$ in $n$ variables. I am interested in the set of bounded polynomials on the sphere $$B_{d,n}=\{f\in V_{...
0
votes
0
answers
121
views
Fundamental primal polynomial associated to an integer
Under Goldbach's conjecture, let $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$ and $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$ for any composite positive integer $n$. Let also $g_{1}(n),\cdots,...
0
votes
1
answer
89
views
Big polynomial subalgebra of polynomials
Consider some algebraically independent polynomials $f_1,\ldots, f_n\in\mathbb{C}[x_1,\ldots, x_n]$.
Is it possible that $I\subseteq\mathbb{C}[f_1,\ldots, f_n]\subsetneq\mathbb{C}[x_1,\ldots, x_n]$ ...
1
vote
1
answer
147
views
Global polynomial basis for the kernel of a matrix polynomial
Let $M(x)$ be an $m$ by $n$ matrix with entries in $\mathbb{C}[x]$. Suppose that for all $x\in \mathbb{C}$ the rank of $M(x)$ is constant and equal to $r<n$. Therefore, for any $x_0\in \mathbb{C}$ ...
0
votes
0
answers
124
views
How to find the polynomials that define a compact Matrix Lie group from its Lie algebra?
Consider a compact (connected) Lie group, or more generally, a linear algebraic Lie group. Suppose we are given the Lie algebra corresponding to the Lie group. How can we find a set of polynomial ...
1
vote
0
answers
75
views
Prove that a translate of a polynomial has roots without using FTA
Let $F\in\mathbf C[Z]$ be a polynomial of degree $d$. Without using the Fundamental Theorem of Algebra, is it true that there is some $u\in\mathbf C$ such that $F + u$ has $d$ roots (counted with ...