2
votes
0answers
113 views

Rewrite sum of radicals equation as polynomial equation

My question is about a method described in Dr.Math forum for simplifying equations involving sums of radical functions. (The following is a transcription of the example given by Dr. Vogler): --- ...
1
vote
2answers
385 views

Given a formal power series ,decide whether there exists a polynomial the series satisfies and if it exists,how to write it down?

Given a formal power series $$y(x)=\sum_{i=0}^{\infty} a_i x^i$$ Is there an algorithm that decides whether there exists a polynomial$$ P(x,y)=p_n(x)y^n+p_{n-1}(x)y^{n-1}+\cdots+p_0(x)=0,p_j(x)\in ...
1
vote
0answers
83 views

Positivstellensatz for non-polynomial term

Can we use Positivstellensatz (P-satz) below for a non-polynomial term? P-satz: Let $R$ be real closed field. Let $f,g,h$ be finite families of polynomials in $R[X_{1} ,...,X_{n}]$. Denote by P the ...
9
votes
1answer
333 views

How is a descent datum the same as a comodule structure?

For a homomorphism of commutative rings $f:R\to S$, there are at least two notions of a descent datum for this map. One of these is to be an $S$-module $M$, with an isomorphism $M\otimes_R S\cong ...
0
votes
0answers
177 views

Monomial ideals: isomorphism problem for commutative algebras?

Theorem 5.27 in Polytopes, Rings, and K-Theory (Bruns, Gubeladze - 2009 - Springer SMM) claims: Let $K$ be a field and $I\!\unlhd\!K[x]= K[x_1,\ldots,x_n]$ and ...
1
vote
1answer
179 views

jacobian polynomial

Here is the question which could be quit difficult (but could be not): Let $C$ be a field of complex numbers and $f \in C[x,y]$ be a polynomial such that there exist $g \in C[x,y]$ and $Jac(f,g) \in ...
-1
votes
1answer
142 views

product of two non-decomposible (closed) polynomials

Let $C$ be the field of complex numbers. Polynomial $f \in K[x,y]$ is called closed or non-decomposible if $C[f]$ is algebraically closed (definition for $f \in K[x1,...,xn]$ is the same). Theorem. ...
1
vote
1answer
170 views

Ring of a Spectral Space

It is said, as far as I can tell that an arbitrary spectral space, i.e. a space that is $T_0$, sober and quasi-compact whose collection of quasi-compact open sets forms a basis and is closed under ...
2
votes
1answer
518 views

Rank of a linear combination of quadratic forms

Suppose we have a set of quadratic forms $Q_i (x_1, \dots, x_n)$ for $1 \leq i \leq k$ in $n$ variables, defined over $\mathbb{R}$. We suppose these are 'collectively nondegenerate' in the sense that ...
17
votes
4answers
2k views

Invariant Polynomials under a Group Action (hidden GIT)

Let's say I start with the polynomial ring in $n$ variables $R = \mathbb{Z}[x_1,...,x_n]$ (in the case at hand I had $\mathbb{C}$ in place of $\mathbb{Z}$). Now the symmetric group $\mathfrak{S}_n$ ...