Sufficient syntactic conditions for zero-dimensionality of polynomial systems 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 $K$).
 A: This is just an elaboration on my comment.
Here are two sufficient conditions:

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*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) says that any set of nonzero polynomials with mutually coprime leading terms is a Gröbner basis. Using this fact, we can easily see that

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*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).
