Abstract
Let k be a field, x = k[x 1, …, x n] the polynomial ring in n variables over k, k(x) the field of fractions of k[x], and L a subfield of k(x) containing k. Hilbert’s fourteenth problem asks whether the k-algebra L ∩ k[x] is finitely generated. The answer to this problem is affirmative if \( \operatorname *{\mathrm{tr.deg}}_kL\le 2\) due to Zariski. In 1958, Nagata gave the first counterexample in the case \( \operatorname *{\mathrm{tr.deg}}_kL=4\) and n = 32. In this chapter, we give a powerful method for constructing counterexamples to Hilbert’s fourteenth problem. Using this method, one can produce various kinds of counterexamples, including those in the case \( \operatorname *{\mathrm{tr.deg}}_kL=n=3\).
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van den Essen, A., Kuroda, S., Crachiola, A.J. (2021). Counterexamples to Hilbert’s Fourteenth Problem. In: Polynomial Automorphisms and the Jacobian Conjecture. Frontiers in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-60535-3_2
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DOI: https://doi.org/10.1007/978-3-030-60535-3_2
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