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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 $K$).

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    $\begingroup$ Doesn't $m=n=2$, $p_1 =x_1 + x_2, p_2 = x_2 + x_1$ satisfy your condition as stated? Maybe you mean something slightly different... $\endgroup$
    – Will Sawin
    Commented Feb 2, 2021 at 14:01
  • $\begingroup$ @WillSawin. Yes, thanks. I have edited to reflect better what I was conjecturing. $\endgroup$
    – Michele
    Commented Feb 2, 2021 at 14:11
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    $\begingroup$ I still don't get it - we can take $c_1 = c_2 =2$, $q_1= x_2/2$, $q_2=x_1/2$. $\endgroup$
    – Will Sawin
    Commented Feb 2, 2021 at 14:13
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    $\begingroup$ One sufficient condition, which probably doesn't apply in your case, is that $p_i$ is a polynomial only in $x_1,\dots, x_i$ and is monic in $x_i$. $\endgroup$
    – Will Sawin
    Commented Feb 2, 2021 at 14:55
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    $\begingroup$ It is a contraction on $\mathbb C[[x]]$, and your argument is completely valid for solutions there (and can be given a more algebraic flavor if desired). The issue is that it's not a contraction on $\mathbb C((x))$ because if the $x_i$ have poles then raising them to powers makes them even larger. So you need a separate argument to handle the case $x_i\notin \mathbb C[[x]]$... $\endgroup$
    – Will Sawin
    Commented Feb 2, 2021 at 17:44

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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) 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).

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