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

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 …

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 $ …

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 …

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 …

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}[ …

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 …

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 …

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 …

(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 …

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 …

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 …

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 …

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 …

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 …