All of the many proofs of the Nullstellensatz I have seen use results from long after Hilbert’s time: Zariski’s lemma, Noether normalization, the Rabinowitch trick, model theory, etc. How did Hilbert’s original proof go?
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$\begingroup$ Duplicate of this stackexchange post. $\endgroup$– Carl-Fredrik Nyberg BroddaCommented Mar 24, 2020 at 17:03
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2$\begingroup$ That MSE link gives proof written in German.. Can one ask for translation to English of Hilbert's proof of Nullstellensatz? $\endgroup$– Praphulla KoushikCommented Mar 24, 2020 at 17:06
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3$\begingroup$ an English transation is available here $\endgroup$– Carlo BeenakkerCommented Mar 24, 2020 at 18:59
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$\begingroup$ Some discussion of the proof of the Nullstellensatz taken from webusers.imj-prg.fr/~michael.harris/theology.pdf pg. 3, footnote 3 which cites 'Algorithms in invariant theory' by Bernd Sturmfels and pg. 7, footnote 7 which states, "These are notably differential methods related to the Lie-Klein theory of continuous groups which later served Hilbert as a framework for invariant theory; and irrational invariants closely tied to his discovery and use of the Nullstellensatz." Of course either of these would be survey in nature, I reckon. But if diving into Hilbert's paper is undesired... $\endgroup$– Samantha YCommented Mar 24, 2020 at 19:41
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$\begingroup$ I should add that that paper is 'Theology and Its Discontents: The Origin Myth of Modern Mathematics' by Colin McLarty and (as the title suggests) about the basis theorem and not the Nullstellensatz. But it came to mind and is a fun and interesting paper. $\endgroup$– Samantha YCommented Mar 24, 2020 at 19:46
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2 Answers
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Here is a scan of the relevant pages from the english translation of Hilbert's 1893 paper.
answered Mar 27, 2020 at 8:17
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$\begingroup$ For reference, these are pp. 234 and 235 of the book. $\endgroup$ Commented Mar 6, 2023 at 8:53
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As noted in math.stackexchange, the theorem is in part 3 of this paper: Hilbert, D. (1893). Über die vollen Invariantensysteme. Math. Ann. 42, pp. 313–373.
From a quick overview, I think the idea is to first prove the theorem when the set of common zeros is finite, using elimination theory. Everything is done with homogeneous polynomials.
Now to find time to read it carefully.
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$\begingroup$ I copied the relevant pages from the english translation of Hilbert's 1893 paper. If there is an interest, drop me a line and I can email these. $\endgroup$ Commented Mar 25, 2020 at 12:59
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$\begingroup$ I would definitely like a cooy but I don’t know how to “drop you a line”. $\endgroup$– FernandoCommented Mar 25, 2020 at 18:10
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$\begingroup$ @Fernando --- I've posted the scan in the answer box, hoping distribution of these few pages qualifies as "fair use". $\endgroup$ Commented Mar 27, 2020 at 8:17