For a finite set of polynomials $p_1, \dots, p_m \in \mathbb{C}[t_1,\dots, t_n]$, is there a deterministic algorithm to determine whether $\langle p_1,\dots, p_m \rangle=\mathbb{C}[t_1,\dots, t_n]$? Regardless of the answer, I would welcome references to work in this vein.
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1You can look for algorithms that compute the reduced Grobner basis of the ideal, though as you may know this approach is rather demanding. The most elementary ways to do it are with variants of Buchberger's algorithm. A fine introduction to the subject is the textbook Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra by Cox, Little and O'Shea. – Apr 18 '20 at 17:18
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Before writing your own code for computing the reduced Grobner basis, try firing up Macauley 2 and see if it succeeds on the polynomials of interest: https://faculty.math.illinois.edu/Macaulay2/TryItOut/ You can try various monomial orders -- some may work when others don't. – Elle Najt Apr 18 '20 at 17:55
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Thanks! My specific question is actually slightly different, and I'm wondering if this matters: I have homogeneous polynomials and want check whether their only common root is 0. This amounts to asking whether $\langle p_1,\dots, p_m \rangle \supseteq I_s$ for some $s \geq 1$, where $I_s$ is the ideal of polynomials of degree $\geq s$. – Ben Apr 18 '20 at 18:12
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Have you tried resultants? For example, $p_1=(x-a)(x-b)$ and $p_2=(x-c)(x-d).$ The resultant eliminating $x$ is $(a - c) (b - c) (a - d) (b - d)$ and this is $0$ iff they have a common root. – Somos Apr 18 '20 at 19:24
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@Somos I have not. This is an excellent suggestion! It seems to be exactly what I'm looking for, and is built in to Macaulay 2: http://www2.macaulay2.com/Macaulay2/doc/Macaulay2-1.13/share/doc/Macaulay2/Resultants/html/_resultant.html – Ben Apr 18 '20 at 19:39
