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)](http://www.ams.org/journals/proc/1996-124-03/S0002-9939-96-02955-3/S0002-9939-96-02955-3.pdf) the authors say there is a classical criterion given by the resultant and refer to chapter I of a book [(link)](http://ia600307.us.archive.org/34/items/algebraictheoryo00macauoft/algebraictheoryo00macauoft.pdf) 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. 



Criteria for system of parameters in polynomial rings 
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 $R\_1R\_2...R\_n \neq 0$.


