Hilbert's fourteenth problem

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In mathematics, Hilbert's fourteenth problem, the fourteenth of Hilbert's problems proposed in 1900, asks whether certain algebras are finitely generated. Let k {\displaystyle k} be a field and let K {\displaystyle K} be a subfield of the field of rational functions k ( x 1 , … , x n ) {\displaystyle k(x_{1},\ldots ,x_{n})} containing k {\displaystyle k} . Hilbert asked whether the k {\displaystyle k} -algebra R := K ∩ k [ x 1 , … , x n ] {\displaystyle R:=K\cap k[x_{1},\ldots ,x_{n}]} is always finitely generated over k {\displaystyle k} . The answer is negative in general, but depends strongly on dimension. Writing r = trdeg k ⁡ K {\displaystyle r=\operatorname {trdeg} _{k}K} for the transcendence degree of K {\displaystyle K} over k {\displaystyle k} , Oscar Zariski (1954) proved that R {\displaystyle R} is finitely generated whenever r ≤ 2 {\displaystyle r\leq 2} ; in particular, the answer is affirmative for n ≤ 2 {\displaystyle n\leq 2} . Masayoshi Nagata constructed the first counterexamples in 1958, one of which he announced at the 1958 International Congress of Mathematicians; his first construction has n = 32 {\displaystyle n=32} and r = 4 {\displaystyle r=4} . In characteristic zero, Shigeru Kuroda subsequently constructed counterexamples with n = 4 {\displaystyle n=4} and r = 3 {\displaystyle r=3} in 2004, and with n = 3 {\displaystyle n=3} in 2005. Thus the bounds are sharp: finite generation always holds when n ≤ 2 {\displaystyle n\leq 2} or r ≤ 2 {\displaystyle r\leq 2} , whereas counterexamples already occur with n = r = 3 {\displaystyle n=r=3} .