Question

Let $k$ be a field and let $R=k[x_1,...,x_n]$ be a polynomial ring over $k$. A subset $\lbrace y_1,...,y_n\rbrace$ of $R$ is called a homogenous system of parameters (hsop) of $R$, if

• the $y_i$ are homogenous polynomials
• $R$ is finitely generated as $k[y_1,...,y_n]$-module

I am looking for criteria, when $\lbrace y_1,...,y_n\rbrace \subseteq R$ is a hsop of $R$.

In the paper (link) the authors say there is a classical criterion given by the resultant and refer to chapter I of a book (link) of Macaulay. Unfortunately I'm not able to figure out what the criterion might be.

Can anyone provide me that criterion or knows alternative criteria ? Any hint is appreciated.

Edit: I want to add that the following turned out to be helpful for working with hsop's over finite fields:

• Hailong's criterion: $\lbrace y_1, ..., y_n \rbrace$ is a hsop $\Leftrightarrow$ $0$ is the only common root of $y_1,...,y_n$ in the algebraic closure of $k$

• Inspecting the proof of Proposition 11.13 in Atiyah-MacDonald's book on commuative algebra shows that a hsop can be obtained in the following way:

  Choose any $y_1 \neq 0$ and if $y_1,...,y_i$ are found, $y_{i+1} \in R$ can be taken to be any polynomial not contained in the minimal primes of $(y_1,...,y_i)$.

For example, I wanted to prove that for a hsop of $\mathbb{F}_p[x_1,...x_n]$ the polynomial functions of the $y_i$ are non-zero. But since $y_1 \neq 0$ can be any polynomial, using Hailong's criterion, it's easy to see that $$y_1 = x^py-xy^p, y_2= x^m + y^m \in \mathbb{F}_p[x,y]\quad m=\frac{p+1}{2}$$ is a counterexample.

Answer 1

Recall that the resultant $Res(g_1,\cdots, g_n)$ of $n$ (not necessarily homogenous) polynomials in $n-1$ variables is $0$ if and only if the $g_i$s have a common root.

Now, the $y_1,\cdots, y_n$ form an sop iff $(0,\cdots, 0)$ is the only common root. But if they have another common root, then because of homogeniety they must have a root with $x_i=1$ for some $i$. That observation leads to the following criterion:

Let $g_{ij}$ be the polynomial $y_i|_{x_j=1}$ and $R_j = Res(g_{1j}, \cdots, g_{nj})$. Then the $y_i$s form an sop iff $R_1R_2...R_n \neq 0$.

Answer 2

See the introduction of the following paper for an answer

http://www.ams.org/journals/proc/2011-139-08/S0002-9939-2011-10790-1/home.html

It cited the paper G. Scheja, U. Storch, Uwe Differentielle Eigenschaften der Lokalisierungen analytischer Algebren, Math. Ann. 197 (1972), 137–170.