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, ...

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2
votes
1answer
70 views

Division of multivariable polynomials by an ideal

Let $K$ be a commutative field and consider an ideal $I$ of $K[X_1,\dots,X_n]$. Is there a well behaved "reduction modulo I", in the following sense : Given a well-ordering $\leq$ on the set of ...
3
votes
1answer
71 views

ideal generated from a truncated “real radical-like” set are still real radical?

Suppose $\mathbb{R}[X]$ is the normal multivariate polynomial ring where $X = x_1...x_n$. $\mathbb{R}[X]_t$ is the truncated ring such that $\mathbb{R}[X]_t =\left\{f: f \in \mathbb{R}[X], deg(f) \leq ...
4
votes
2answers
92 views

Is there a critical point of a polynomial $f$ within every disc having as diameter the line segment between two zeros of $f$?

Consider a generic complex polynomial $f$ (i.e. one with pairwise distinct zeros [EDITed:] and with pairwise distinct zeros of $f'$) and define for every pair of zeros an open disc which has as ...
2
votes
0answers
165 views

Hilbert series of the weight 0 sub-algebra of the algebra of functions on GL(N)

Let $A$ be the algebra of polynomials in $N^2$ variables $x^i_j$, $i,j=1,\dots,N$. It is $\mathbb{Z}^N$ graded, with $\text{weight}(x^i_j)=e_i-e_j$. Here $(e_1,\dots,e_n)$ is the standard basis of ...
1
vote
0answers
89 views

approximation of rational functions

Suppose $\hat{p}/\hat{q}$ and $p/q$ are two rational functions where $p,q,\hat{p},\hat{q}$ are of degree $n$. Suppose they satisfy that $|p(z)/q(z) - \hat{p}(z)/\hat{q}(z)| < \epsilon$ for any $z$ ...
0
votes
1answer
95 views

Functions of several variables over finite fields [closed]

For a finite field $F$ any function $f\colon F\to F$ is given by a polynomial. My question is what happens when we are given a function of two or more variables? Is this necessarily a polynomial ...
7
votes
0answers
132 views

Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)?

The OEIS entry A121448 enumerates binary trees with $n$ edges and $k$ vertices with outdegree 1. Has anyone seen these trees? The o.g.f. for this entry, $G(x,t)$, is essentially a discriminant ...
1
vote
0answers
96 views

Boundary of pseudospectra

Suppose: $B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$ ${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...
1
vote
1answer
203 views

Methods for searching for prime generating polynomials

I am currently using a non-systematic, pseudo-random method for finding prime-generating polynomials, based on the Bateman-Horn method for finding likely candidates, and then narrowing down. I have ...
0
votes
1answer
88 views

approximation of products of polynomials

I am wondering whether the following can be proved: Suppose $p(z)$ and $q(z)$ are polynomials of degree $n$ with real coefficients and leading coefficient 1. Moreover, they have quite different ...
1
vote
1answer
113 views

When is a homogeneous polynomial an inner function on the unit torus?

What is the necessary and sufficient condition(s) for a homogeneous polynomial on torus(bidisk) to be an inner function? More precisely I want to know when absolute value of a homogeneous polynomial ...
5
votes
2answers
125 views

Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian

This is a question related to the statistical model behind independent component analysis (ICA). We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a ...
0
votes
0answers
35 views

finite orbits of transformations of the rational function field

Let $K : = k(x_1, x_2, \cdots, x_t)$ be the rational function field and consider the transformation $\tau$ of $K$ defined by $u \mapsto \frac{\alpha(u) + b}{a}$, where $a \ne 0 , b \in k$ and $\alpha$ ...
0
votes
0answers
50 views

Is there any way to approximate the largest root of a polynomial? [duplicate]

I know Newton's method will always give a root, but is there a modified version of the method that will always give the largest root?
1
vote
1answer
120 views

Is there a limit definition for the roots of a polynomial with arbitrary degree? [closed]

I know there's no general formula for all the roots of a polynomial with a degree greater than 4, but is there some sort of limit (or other) definition to calculate the roots (particularly the largest ...
2
votes
0answers
123 views

Polynomials with some roots whose product is 1

I asked this question in this post but have not got a full answer. So I post it again on MO. Consider the complex coefficient polynomial equation \begin{eqnarray} ...
3
votes
1answer
194 views

Polynomial factoring over finite fields

What is known in general about the complexity of factoring polynomials over finite fields? For instance given $\Bbb F_q$ where $q=p^n$ and total degree $d$ polynomial in $m$ variables what can we say ...
30
votes
3answers
1k views

Polynomials with the same values set on the unit circle

Assume that $P(z)$, $Q(z)$ are complex polynomials such that $P(S)=Q(S)$, where $S=\{z\colon |z|=1\}$ (equality is understood in the sense of sets, but I do not know the answer even for multisets). ...
1
vote
1answer
94 views

Ternary cyclotomic polynomials with $n=15r$

Let $n=15r$ where $r>5$ is an odd prime number. If $r\!\!\! \mod 15 \equiv w$ then is it true that $\Phi_{n}(x)$ is not flat whenever $2<w<13$ ? In other words, are the flat ones necessarily ...
2
votes
0answers
95 views

Log-concave polynomial is a log-concave function?

A polynomial $\sum\limits_{k=0}^n a_kx^k$ is log-concave if $a_0,\ldots,a_n$ constitute a log-concave sequence. I wonder whether the log-concave polynomial is also a log-concave function with respect ...
0
votes
0answers
15 views

Complexity of computing the multivariate Tutte polynomial of clique where each edge have distinct label

The multivariate Tutte polynomial $Z_G(q,v)$ is generalization of the Tutte polynomial and each edge is labelled by variable $v_e$. $Z_G(q,v)$ is linear in $v_i$. Let $G$ be a clique where each edge ...
1
vote
0answers
20 views

Complexity of computing the Tutte polynomial of multigraph when the Tutte polynomial of the underlying simple graph is known

Let $G$ be multigraph with $l$ loops and $m$ multiple edges and $G'$ be the underlying simple graph (loops and multiple edges removed). Assume the Tutte polynomial of $G'$ is given. Q1 What is ...
5
votes
2answers
148 views

Generalized cycle index polynomial for the symmetric group

The answer to a particular calculation in quantum information theory gives me the following expression: Given $M$ specific elements of the symmetric group $S_n$, define the polynomial $$Z_n(\pi_1, ...
4
votes
0answers
158 views

Polynomials representing locally constant functions

Let $K$ be a finite field with $p$ elements. (a) Let $f\in K\lbrack x\rbrack$ be such that (i) $\deg(f)<p$ and (ii) $f(2x) = f(x)$ for $\geq (1-\epsilon) p$ values of $x$ in $K$. What can we say ...
2
votes
1answer
207 views

On the divisibility of a certain power sum

Does $1^n + 2^n + \cdots + m^n$ divide $(1+2+ \cdots +m)^n$ for any even integers $m, n\geq 2$ ?. For $n\leq 4$, the solution easily follows from the relevant identities. For $n\geq 6$, i suspect ...
5
votes
0answers
156 views

Is a wild automorphism of $k[x_1,\ldots,x_n]$, $n \geq 3$, necessarily of infinite order?

Let $k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$ of characteristic zero. When $n=2$, it is known that every automorphism of $k[x_1,x_2]$ is tame, namely, a finite product of elementary ...
0
votes
1answer
115 views

Standard rational functions from matrices

In linear algebra we get introduced to standard polynomials that are associated to matrices such as characteristic polynomials and determinants. What are some of the standard rational functions that ...
0
votes
0answers
45 views

Approximation by polynomials with coefficients sum

In this paper Erdos p.1176 remarked that if the coefficients of $f(z):\sum_{v=0}^{n}a_{v}z^v$ are all real,then $\sum_{v=0}^{n}|a_{v}|$ is maximal for $f(z)=\pm T_{n}(z)$,where $T_{n}(z)$ is Chebyshev ...
2
votes
1answer
75 views

Distribution of values of quadratic polynomials over a finite field

Let $S$ denote the set of squares in the finite field ${\mathbb F}_p$. Now let $f(x) \in {\mathbb F}_p[x]$ denote a degree $2$ polynomial, which is not a square of a linear polynomial. If we consider ...
0
votes
2answers
404 views

Solving a system of equations using Gröbner basis

In Sage (or any other package) when using Gröbner basis to solve a system of equations (some of which are non-linear equations) does computing the Gröbner basis for the ideal ID generated by the ...
2
votes
0answers
95 views

system of complex number equations

Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that $$a_1^3+a_2^3+a_3^3+a_4^3=0$$ $$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$ ...
5
votes
1answer
210 views

Estimating size of greatest prime divisor of a sequence of integers

Consider the numbers of the form: $$A_n = \prod_{\pm}\left(\pm 1\pm \sqrt{2} \pm \cdots \pm \sqrt{n}\right)$$ where, the product in taken oven all $2^n$ terms with variations in sign. We know such ...
1
vote
0answers
60 views

Octahedron and System of trigonometric equations

Could somebody help me to prove the following? $$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \cos (\phi_k)=0$$ ...
-3
votes
1answer
204 views

Does differentiation widen, or narrow, the class of functions?

Let $\cal F^k$ be a set of functions, each of class $C^k$, i.e., both, for every function in $\cal F^k$: $k^{\textrm{th}}$ derivatives exist, and are continuous. Let $D(\cal F^k)$ be the set of ...
3
votes
1answer
154 views

Hermite-Kakeya Theorem for entire functions

In a question asked by Bobby Ocean, the following theorem is cited: Hermite-Kakeya Theorem(for polynomials) - Given two real-valued polynomials, $f$ and $g$, then $f(x)+g(x) r$ has only real zeros ...
1
vote
1answer
171 views

Reduced resultant of monic polynomials

Let $f(x)$ and $g(x)$ be coprime monic polynomials in $\mathbf{Z}[X]$ of positive degrees $m$ and $n$ respectively. It seems that in this case their reduced resultant can be obtained from the ...
8
votes
1answer
568 views

Sort-of Converse of Kolmogorov Zero-One Theorem

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. The Kolmogorov Zero-One Theorem states that Suppose we have independent random variables $X_1, X_2, ...$. Then $\forall \ A \in ...
11
votes
2answers
374 views

regular polygon question

Let $a_1,a_2,\ldots,a_n$ be distinct points on the complex plane $\mathbb{C}$ and $L$ be a circle in $\mathbb{C}$ such that $$f(z):=\sum_{i=1}^n|z-a_i|^{2n-2}$$ is constant on $L.$ Could somebody ...
2
votes
2answers
104 views

Quick tests to differentiate eigenvalues

Given two real square symmetric matrices $A$ and $B$ are there any quick tests to make sure at least one of their eigenvalues differ without computing the eigenvalues and likely more robust or looking ...
3
votes
0answers
45 views

numerical stability of root identification via Newton-Raphson iteration of Stieltjes residue sums

I have asked several questions on math.SE in order to compute numerically the poles of high-degree Padé approximations for $e^{-x}$, because a computation directly from the polynomial ...
0
votes
0answers
67 views

On a (possible?) equivalence of Bunyakovsky conjecture

Dirichlet's Theorem on primes in arithmetical progressions is equivalent to the following statement: for all integer numbers $a,b$, where $\gcd(a,b)=1$, there exists at least one prime of the form ...
2
votes
0answers
112 views

Zeros of polynomials modulo non-prime

Suppose I have a set S and I want to find a polynomial p such that $p(s) = 0 \mod n$ if $s \in S$, and that it is non-zero modulo n otherwise. In the literature such an S is sometimes called a root ...
4
votes
0answers
80 views

symmetric systems of polynomial equations

Suppose I have a polynomial $p(x_1,...,x_N)$ in $N$ complex variables, and I wish to solve $p(x_{\pi(1)},...,x_{\pi(N)})=0$ for all permutations $\pi \in S_N$. Clearly this is overdetermined for ...
2
votes
1answer
88 views

Pointwise convergence of polynomials to a function on a compact set K that is 1 on some disc D and zero outside D

Motivation of my question: Let $A$ be a bounded selfadjoint operator with spectral measure $E$ and $x$ a vector. Then it is easily seen that the closed linear span of all $A^nx$ ($n\in\mathbb N$) ...
3
votes
0answers
277 views

A relation between a ring with its polynomial ring

Let $\{f_i(x)\}_{i\in I}$ be a subset of $R[x]$ where $R[x]$ is the polynomial ring of $R$(a commutative ring with identity). If the ring $R/\langle f_i^2(n)-f_i(n)\rangle_{i\in I, n\in A}$, for every ...
0
votes
2answers
86 views

Number of polynomial terms for certain degree and certain number of variables

I would like to calculate the maximum number of polynomial terms given a certain number of variables and a certain degree. eg. given that the number of variables is 2 and the degree is 3, the maximum ...
5
votes
2answers
98 views

(Re)construction of a polygon from all inter-vertex distances

For a $n$-polygon in $\mathbb{R}^3$ the set of distances between all pairs of vertices is given. (How) is it possible to reconstruct the geometric structure of the polygone? Symbolically: For a set ...
0
votes
2answers
487 views

What is wrong with this counterexample to the Weak Bunyakovsky's conjecture and reformulation of Bunyakovsky's conjecture?

From HYPOTHESIS H AND AN IMPOSSIBILITY THEOREM OF RAM MURTY. On p. 13 BUNYAKOVSKY’S CONJECTURE ( WEAK FORM ). Let $f$ be a polynomial with integer coefficients and positive leading coefficients ...
4
votes
0answers
157 views

Plane real curves such that their intersections with lines are hyperbolic

Let $R$ be an (irreducible) plane real algebraic curve (without isolated points). Consider Zarissky closure of $R$ in ${\mathbb C}P^1$ (as real variety). Suppose that $\lambda\in R ...
10
votes
0answers
188 views

Precise relationship between “finite” Fourier analysis and Galois theory in solving the cubic?

Suppose we want to solve$$x^3 - ax^2 + bx - c = 0.$$We know a priori that this can be factored as $(x - r_0)(x - r_1)(x - r_2)$; by Vieta's formulas, we know$$a = r_0 + r_1 + r_2,\quad b = r_0r_1 + ...