ac.commutative algebra - Filtration over tensor product

groebner bases – Methods for multivariate polynomial equations over sizable finite fields retort

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groebner bases – Methods for multivariate polynomial equations over sizable finite fields

I’m making an attempt to amass a harsh overview of one of the best strategies one power make use of to search out options of multivariate polynomial equations over sizable finite fields. We can suppose for simplicity that the given system has a finite variety of options (over the closure).

As I win it, the widespread algebraic strategies for locating options over rational numbers are both based mostly on the Grobner foundation (computing the elimination mannequin) or are based mostly on the resultants. First query: Are there every other widespread algebraic strategies?

Second (most important) query: Is the standing over finite fields any totally different? Are there every other widespread strategies which solely labor for finite fields?

To perhaps illustrate my confusion from all of the sources I maintain been going via:
The undertaking on web page 3 of this thesis on finite bailiwick multivariate polynomial equations is making an attempt to present the reader some harsh overview. But it does not comprise any resultant-based strategies. Are these not apt for finite fields for some instinct I’m not seeing?

Any solutions or sources to solutions would breathe very appreciated.

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