-1
votes
0answers
44 views
Solution of Equation [closed]
Can anyone show me, how to solve these system of Equations:
x+y+z = 2
(x+y)(y+z)+(y+z)(z+x)+(z+x)(x+y) = 1
X^2(y+z)+Y^2(z+x)+Z^2(x+y) = -6
3
votes
0answers
91 views
identity for number of monomials
Fix a positive integer $d \geq 2$, and let $n,k$ be natural numbers with $k \leq n$.
Let $b(n,k)$ denote the number of monomials of degree $kd-(n+1)$ in $n+1$ variables $x_0,\ldo …
3
votes
1answer
210 views
Algebraic closure of a polynomial ring
What could be conditions on $k\in\mathbb{C}[x,y,z]$ that would ensure that any polynomial $f\in\mathbb{C}[x,y,z]$ that is algebraically dependent of $k$ is indeed a polynomial in $ …
2
votes
1answer
74 views
Multivariate polynomial approximation of smooth functions
Let $f$ be a function defined on $[-1,1]^d$. Assume that all partial derivatives of $f$ up to order $r$ are continuous; and the $\infty$-norm of these partial derivatives are unifo …
0
votes
1answer
55 views
Existence of nontrivial roots of a homogeneous polynomial over a finite field in extension fields
Consider a homogeneous polynomial, f, of total degree n in n variables, with coefficients in a prime order finite field, GF(p).
Are there any general rules regarding the existence …
1
vote
1answer
108 views
How to show an ideal is Zero-dimensional [closed]
I have the following past exam paper question, a similar sort of question seems to come up every year.. And i'm completely lost with it...
Let $J$ denote the ideal in $\mathbb{Q}[ …
0
votes
1answer
99 views
Zeros of compositions of polynomials and derivatives
Suppose we have polynomials $f$ and $g$ over a field $F$ and for some $a\in F$ we know $(x-a)^m$ divides $f\circ g$, i.e. $f\circ g$ has a zero of order $m$ at $a$. Then we have th …
3
votes
2answers
360 views
A basis of the symmetric power consisting of powers
I have asked this question on math.se, but did not get an answer - I was quite surprised because I thought that lots of people must have though about this before:
Let $V$ be a com …
0
votes
0answers
51 views
When do the even part and odd part of a hypergeometric like function have common nonnegative real root?
I encountered the following problem in my research project:
Let
$$f(x)=\sum_{k=0}^{\infty} a_{k} x^{k}$$
We can separate the even part $p(x^2)$ from the odd part $x q(x^2)$ and …
6
votes
3answers
325 views
What is the correspondence between combinatorial problems and the location of the zeroes of polynomials called?
(From MSE)
In the wikipedia article on the Italian-born American mathematician and philosopher Gian-Carlo Rota, it is stated that the one combinatorial idea he would like to be re …
1
vote
0answers
216 views
When integer polynomials take integer values, does their GCD also take integer values?
Suppose you have two polynomials $P$ and $Q$ with integer coefficients. Let their GCD (more specifically, the smallest integer multiple of the GCD in $\mathbb Q$ such that all the …
19
votes
1answer
776 views
Have we ever proved any non-solvable case of reciprocity without the Langlands program ?
The reciprocity of the title is the following not completely well-posed problem:
Fix $P(X)$ a monic irreducible polynomial of degree $n$, with coefficients in $\mathbb Z$. "Descri …
2
votes
1answer
163 views
Why can an infinite-order polynomial be written as the ratio of two finite-order polynomials?
I am studying GARCH processes in Time Series Analysis by Hamilton.
Something that has regularly been used in the book is the assumption that an infinite-order polynomial can be wr …
2
votes
1answer
248 views
Is there an algorithm to decide if an ideal contains monomials?
Let $I\subset k[x_1,\dots,x_n]$ be an ideal in a polynomial ring in commuting variables. Is there a procedure to decide if $I$ contains a monomial and possibly to find one?
Gröbn …
25
votes
1answer
533 views
How many polynomial Morse functions on the sphere?
Let $f$ be a homogeneous polynomial of degree $d$ in $n$ variables. Restricted to the unit sphere $S^{n-1}$, it might or might not be a Morse function.
If $f$ is a Morse function …

