**10**

votes

**0**answers

153 views

### The sum of squared logarithms conjecture

I am searching for the first proof of (or counterexample to) the following conjecture.
(The sum of squared logarithms conjecture)
For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , ...

**1**

vote

**0**answers

15 views

### Multivariable polynomial interpolation via evaluations from entrywise powers of a point

I am interested in multivariate polynomial interpolation. Within computational complexity theory, I use it to create efficient reductions between counting problems. In the univariate case, there is ...

**0**

votes

**1**answer

57 views

### When can two Cauchy transforms intersect?

Given two polynomials $p$ and $q$ over reals and being guaranteed that both have all roots real I want to know if there is any characterization of the solutions of the equation $\frac{p'}{p} = ...

**0**

votes

**0**answers

97 views

### compute standard basis in local rings

Let$>'$ be the order in $k[t,x_1,\cdots,x_n]$ as follows:
Each semigroup order > on monomial in the $x_i$ extends to a semigroup order >' on monomial in $t,x_1,\cdots,x_n$ in the following way. We ...

**0**

votes

**0**answers

80 views

### Polynomial generated with primitive element modulo p

This question is equivalent to the question "Normal basis in cyclotomic number fields" that I asked recently. I am posing this question because maybe in this format somebody can have an answer:
Let ...

**0**

votes

**0**answers

51 views

### Find all possible rational pairs of a parametric sextic and all cases where it is reducible for either parameter

Find all possible rational solutions pairs $(z,a)$ of the equation
$a^6 + a^4 (-18368 + 9184 z - 2912 z^2) + a^2 (61702144 - 61702144 z + 36814848 z^2 - 10694656 z^3 + 1748992 z^4) - z^2 ...

**0**

votes

**0**answers

12 views

### Is there a field in which every rational polynomial has a root (other than the obvious fields)? [migrated]

Let $\mathbb{A} \subset \mathbb{C}$ denote the field of numbers algebraic over $\mathbb{Q}$. Is there a proper subfield $F$ of $\mathbb{A}$ such that every nonconstant polynomial $p(x) \in ...

**0**

votes

**0**answers

39 views

### A constrained positive polynomial

Is there an example of a polynomial $Q(x)\in\Bbb Z_{\geq0}[x]$ with $Q(0)=1$ so that $Q(x)=Q_m(x)Q_+(x)$ where $$Q_+(x)\in\Bbb Z_{\geq0}[x]$$ $$Q_m(x)\in\Bbb Z[x]\mbox{ so that }Q_m(x)\mbox{ has at ...

**4**

votes

**0**answers

214 views

### Analog of Euler's factoring technique

Is there an analog of Euler's Two Squares factoring theorem over polynomial rings $\Bbb Z[x]$ by considering a version for non-negative polynomials?
Euler's two squares factoring states that numbers ...

**13**

votes

**6**answers

2k views

### On Polynomials dividing Exponentials

EDIT, May 2015: in the second edition of the relevant book, the question was corrected, a single number had been mis-typed. The corrected question (thanks to Max) is to find all positive integer pairs ...

**2**

votes

**0**answers

102 views

### Given a multivariate polynomial with even degree, can we find its tightest convex polynomial 'envelop'?

To be specific, given a multivariate polynomial function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ with even degree $2d$, can we construct a convex polynomial function $g$, such that:
$\forall ...

**5**

votes

**2**answers

171 views

### Continuous map from connected subset of R^n to one of the real zero of an odd degree polynomial whose coefficients are polynoms of the variables

Let take a real multivariate polynomial $P(x_1, \ldots, x_n, y)$ such as the degree of P relatively to the variable $y$ is odd. Thus, for each $X = x_1,\ldots,x_n \in\mathbb{R}^n$, the univariate ...

**1**

vote

**1**answer

33 views

### Splines linearly independent

Let $N_1:=\chi_{[0,1]}$ be defined as this characteristic function and $N_n:=N_{n-1}*N_1$ then this leads to polynomials with support $[0,n]$. These splines are well-studied click for wikipedia My ...

**7**

votes

**1**answer

1k views

### Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants

Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety; the Artin-Mazur zeta ...

**4**

votes

**1**answer

166 views

### Macaulay's example of prime ideals in $\mathbb C[X_1,X_2,X_3]$ having large number of generators

There is a famous example of Macaulay which shows that there are prime ideals of height two in $\mathbb C[X_1,X_2,X_3]$ having at least $l$ generators for any $l\ge 3$.
In Macaulay's words, the ...

**8**

votes

**3**answers

528 views

### $S_k(x+y)-S_k(x)-S_k(y)$ where $S_k$ is symmetric polynomial

Let $S_k$ be the $k$-th elementary symmetric polynomial of $n$ variables. How can I rewrite $$S_k(x+y)-S_k(x)-S_k(y)$$ by just using $x,y,S_1,S_2,\cdots S_{k-1}$ where $x=(x_1,x_2,\cdots,x_n)$ and ...

**5**

votes

**0**answers

153 views

### Degree of Chebyshev polynomial necessary

In general, given $0<a<1$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[1-\frac{a}2,1+\frac{a}2]$ at every $x\in[1-a,1+a]$ and $f(0)=0$. What is minimum degree that is ...

**11**

votes

**1**answer

268 views

### Orthogonal polynomial under linear transformation

Let $M_n(x) = x^n$ be the standard monomials. The binomial formula allows one to expand $M_n(ax+b)$ as a linear combination of $M_k(x)$, for $k \leq n$, giving
$$
M_n(ax+b) = (ax+b)^n = \sum_{k=0}^n ...

**-1**

votes

**0**answers

47 views

### Must a multivariate polynomial be zero subject to constraint on the values of all pairs of distinct variables at solutions?

I suppose this is both easy and false.
Let $f \in \mathbb{F}_2[x_1, \ldots x_n]$.
Suppose in all solutions of $f(x_1, \ldots x_n)=0$
all pairs of variables $(x_i,x_j),i \ne j$ can take all
possible ...

**1**

vote

**1**answer

159 views

### a problem about ideals of polynomial rings

Let $\{f_n\}_{n=1}^\infty\in \mathbb{C}[x,y]$ be a sequence of polynomials given by the following expressions
$$
f_n(x,y)=\sum_{i=0}^{[\dfrac{n}{2}]}(-1)^{n-i}{{n-i}\choose i}x^{n-2i}y^i.
$$
Let ...

**1**

vote

**1**answer

196 views

### Composition of rational functions

Given a rational function $R\in\Bbb R(x_1,\dots,x_n)$ with multilinear numerator and denominator, is there always a rational function $G\in\Bbb R(x)$ such that $G\circ R\in\Bbb R[x_1,\dots,x_n]$, ...

**1**

vote

**1**answer

38 views

### Dimension of a certain subspace of univariate polynomials

Let $\mathbb{F}$ be an arbitrary field. For a polynomial $f\in\mathbb{F}[x]$,
we use $Z(f)$ to denote set of roots of $f$ in $\mathbb{F}$. Let $S$
and $T$ be sets of elements of $\mathbb{F}$ of size ...

**2**

votes

**0**answers

101 views

### What are the general zonal spherical functions for ${\rm SO}(n)/{\rm SO}(n-1)$?

The zonal spherical functions [1] on the sphere $(G={\rm SO}(n)$, $K={\rm SO}(n-1))$ are the Gegenbauer or ultraspherical polynomials if one considers the irreducible representations of ${\rm SO}(n)$ ...

**41**

votes

**6**answers

2k views

### How to recognise that the polynomial method might work

A couple of days ago I was at a nice seminar given by Christian Reiher, during which he told us about a short proof of the following special case of a theorem of Olson.
Theorem. Let ...

**0**

votes

**1**answer

269 views

### cup-length of the first Chern class of complex grassmannian

Let $G_2(\mathbb{C}^{n+1})$ be the complex grassmannian.
Then the cohomology ring $H^*(G_2(\mathbb{C}^{n+1});\mathbb{C})=\mathbb{C}[c_1,c_2]/(f_n,f_{n+1})$, where ...

**8**

votes

**3**answers

448 views

### Degree necessary of a polynomial?

Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...

**0**

votes

**0**answers

48 views

### A multivariate polynomial question

Split $\{0,1\}^n$ into $S_0[n],S_1[n]$ with $S_0[n]\cup S_1[n]=\{0,1\}^n$ while $S_0[n]\cap S_1[n]=\emptyset$.
For every $n$, let $f(x_1,\dots,x_n),g(x_1,\dots,x_n)\in\Bbb R[x_1,\dots,x_n]$ be ...

**1**

vote

**0**answers

72 views

### Bounds on degree from bounds on derivatives

Let $f(x)\in \Bbb R[x]$ and $r(x)\in\Bbb R(x)$. Supposing we have information about the values taken by $f(x)$ and $g(x)$ in certain intervals and also can bound their derivatives in these intervals, ...

**6**

votes

**0**answers

399 views

### Find a polynomial not in any ideal generated by polynomials of total degree $o(n)$

Is there an explicit nontrivial (= not a constant) polynomial $p \in \mathbb{C}[x_1, \ldots, x_n]$ such that, for any ideal $I \not= \mathbb{C}[x_1, \ldots, x_n]$ generated by $f_1, f_2, \ldots, f_m$ ...

**13**

votes

**1**answer

355 views

### Positive roots of a polynomial

Let $a_i>0$, $i=1,\dots,n$, and put $\overline{a}:=\frac{1}{n}\sum_{i=1}^n a_i$. Assuming not all $a_i$'s are equal, take
$$
p(x):=\sum_{i=1}^n a_i (a_i-\overline{a})\prod_{k=1,\dots,n\;k\neq i} ...

**2**

votes

**1**answer

176 views

### Smallest degree of approximating polynomial

Let $\{0,1\}^n=S_0\cup S_1$ withh $S_0\cap S_1=\emptyset$.
Let $\epsilon\in[\frac{1}2,1)$.
Let $f:\Bbb R^n\rightarrow\Bbb R$ be a polynomial such that $$f(S_0)=0,\mbox{ ...

**7**

votes

**2**answers

164 views

### Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial

The Gauss--Lucas Theorem states that all zeros of a degree $n$ complex polynomial $p(z)$ are contained in the convex hull of the zeros of $p$. By iteration, this implies that the zeros of ...

**2**

votes

**0**answers

109 views

### A multivariable polynomial degree question

Given $\mathsf{F,G}\in\Bbb R[x_1,\dots,x_n]$, minimum multivariate polynomials of least total degree $\mathsf{degF}$, $\mathsf{degG}$ such that, given unequal $a,b\in\Bbb R$,
$$\mathsf{F(p)}=a, ...

**5**

votes

**1**answer

151 views

### Computing minimal polynomials using LLL

I would like to use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL algorithm) to compute the minimal polynomial of a (real) algebraic number $\alpha$ from a decimal approximation ...

**13**

votes

**4**answers

736 views

### Random Diophantine polynomials: Percent solvable?

Suppose one generates a random polynomial
of degree $d$ with integer coefficients
uniformly distributed within $[-c_\max,c_\max]$.
For example, for
$d=8$, $|c_\max|=100$, here is one random ...

**0**

votes

**1**answer

1k views

### How are taps proven to work for LFSRs?

Obviously, you can exhaustively check that it lands on every state except the zero state, but for large linear feedback shift registers (LFSR), this quickly becomes infeasible.
Wikipedia states the ...

**0**

votes

**2**answers

116 views

### Irreducibles in polynomial rings

Let R be a reduced ring with characteristic zero which is not an integral domain. Is "x" necessarily non irreducible in R[x]?

**2**

votes

**2**answers

110 views

### Integrating a barycentric monomial over a simplex

Are there standard formulas for the integral over a simplex of a monomial in the barycentric coordinates? Can someone supply a reference? I think I have seen such formulas, but I am unable to find ...

**3**

votes

**3**answers

364 views

### Polynomials of low degree that clone polynomials of higher degree

Let $f(x_1,\dots,x_{16})=(x_1+x_2+x_3+x_4)(x_5+x_6+x_7+x_8)(x_9+x_{10}+x_{11}+x_{12})(x_{13}+x_{14}+x_{15}+x_{16})\in\Bbb R[x]$.
Let $\mathcal{Z}$ be the zero set of $f$ in ...

**4**

votes

**1**answer

165 views

### Multivariable function analysis

Sorry if the terms I'm going to use is not professional enough:) This is about the complexity analysis of an algorithm.
Let $\alpha$ be the greatest real root of the polynomial ...

**0**

votes

**0**answers

69 views

### Factoring a polynomial in a specific manner

Let $f(x,y,z) = ax^d + b y^k z^{d-k} + c$, where $a,b,c \in \mathbb{C}$ and $abc \ne 0$, and $d \geq 3$. Let $d'$ denote the smallest non-negative integer $l$ such that there does not exist a positive ...

**1**

vote

**0**answers

80 views

### Generalization of Schur polynomials

I am making a list of generalizations of Schur polynomials and other closely related polynomials that appear in representation theory. My motivation is to eventually make a nice poster of ...

**2**

votes

**3**answers

189 views

### ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$.
Let $f,g\in \mathbb{C}[x,y]$.
Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$.
...

**6**

votes

**7**answers

671 views

### Source for roots of matrix polynomials?

A
matrix polynomial
is a polynomial whose variables are square $n \times n$ matrices,
let's say with entries in $\mathbb{C}$, and with coefficients in $\mathbb{C}$.
I am seeking a source of results on ...

**1**

vote

**1**answer

54 views

### ideals of polynomial ring of two variables generated by two elements

Let $f,g$ be two polynomials in $\mathbb{Z}[x,y]$, given by
$$
f(x,y)=x^4-3xy+y^2,$$
$$
g(x,y)=x^5-4xy+3xy^2.$$
Let $I=(f,g)$ be the ideal in $\mathbb{Z}[x,y]$ generated by $f$ and $g$.
Is ...

**5**

votes

**2**answers

166 views

### Common roots of polynomial and its derivative

Suppose $f$ is a uni-variate polynomial of degree at most $2k-1$ for some integer $k\geq1$. Let $f^{(m)}$ denote the $m$-th derivative of $f$. If $f$ and $f^{(m)}$ have $k$ distinct common roots ...

**3**

votes

**0**answers

315 views

### A question on partial fraction decompositions

This question concerns the mapping from the poles of a rational function to the coefficients of its partial fraction decomposition. In general, this mapping is not injective. I want to identify some ...

**1**

vote

**2**answers

234 views

### Ideals generated by two elements in the polynomial ring of two variables over a field

Let $k$ be a field. For example, $k=\mathbb{Q}$ or $\mathbb{Z}/p$, $p$ prime.
Let $k[x,y]$ be the polynomial ring.
Let $f,g\in k[x,y]$.
Let the ideal $I=(f,g)$ be the ideal of $k[x,y]$ ...

**1**

vote

**0**answers

247 views

### Total degree of a polynomial

Let $\mathsf{F,G}\in\Bbb R[x_1,\dots,x_n]$ be minimum multivariate polynomials of least total degree $\mathsf{degF}$, $\mathsf{degG}$ such that, given unequal $a,b\in\Bbb R$,
$$\mathsf{F(p)}=a, ...

**1**

vote

**1**answer

43 views

### local bernstein type inequality for multivariate polynomials

Let's say $p(x_1,...,x_n)$ is an n-variate degree d homogenous polynomial. Assume $U \subset S^{n-1}$ and $ vol(U) > 0 $ is there any Bernstein type inequality saying
$$ \max_{x \in U , y \in ...