Consider a system $S$ of polynomial equations, $p\_1=0,...,p\_m=0$, for $p\_i\in K[x\_1,...,x\_n]$, for a field $K$: the system $S$ is *zero-dimensional* if it has finitely many solutions. It is well-known that zero-dimensionality can be decided by algorithms that rely on the construction of a Groebner basis for the ideal generated by the $m$ polynomials.


My question is, if *sufficient*, nontrivial syntactic conditions on $S$ are known by which zero-dimensionality can be so to speak "read off" from the polynomials in $S$, in particular without having to compute a Groebner basis for it.


(If helpful, restrict to 0 characteristic, algebraically closed fields $K$).



Sufficient syntactic conditions for zero-dimensionality of polynomial systems
This is just an elaboration on my comment.


Here are two sufficient conditions:


* *Condition 1:* We have $m \geq n$, and each polynomial $p\_i$ with $i \leq n$ has the form $p\_i = x\_i^{m\_i} + \left(\text{some polynomial of degree $< m\_i$}\right)$ for some nonnegative integer $m\_i$.
* *Condition 2:* We have $m \geq n$, and each polynomial $p\_i$ with $i \leq n$ has the form $p\_i = x\_i^{m\_i} + \left(\text{some polynomial in the variables $x\_{i+1}, x\_{i+2}, \ldots, x\_n$}\right)$ for some positive integer $m\_i$.


Indeed, Buchberger's first criterion (Theorem 1.1.34 and Lemma 1.1.39 in [Willem de Graaf, *Computational Algebra*, 2019-09-06](http://www.science.unitn.it/%7Edegraaf/compalg/notes.pdf)) says that any set of nonzero polynomials with mutually coprime leading terms is a Gröbner basis. Using this fact, we can easily see that


* Condition 1 is sufficient, because it ensures that $\left(p\_1, p\_2, \ldots, p\_n\right)$ is a Gröbner basis of the ideal generated by $p\_1, p\_2, \ldots, p\_n$ with respect to the deg-lex order (which then entails that this ideal has finite codimension, whence the ideal generated by $p\_1, p\_2, \ldots, p\_m$ has finite codimension a fortiori).
* Condition 2 is sufficient, because it ensures that $\left(p\_1, p\_2, \ldots, p\_n\right)$ is a Gröbner basis of the ideal generated by $p\_1, p\_2, \ldots, p\_n$ with respect to the lex order (which then entails that this ideal has finite codimension, whence the ideal generated by $p\_1, p\_2, \ldots, p\_m$ has finite codimension a fortiori).


