Masayoshi Nagata

Masayoshi Nagata ( Japanese 永田 雅 宜 , Nagata Masayoshi ; born February 9, 1927 in Aichi Prefecture ; † August 27, 2008 in Kyoto ) was a Japanese mathematician who is known for his work in commutative algebra and algebraic geometry . He was a professor at Kyoto University .

Career

Nagata graduated from Nagoya University in 1950. He was a student of Tadasi Nakayama there and assistant after graduation. He then became a lecturer and from 1963 professor at Kyoto University, where he retired in 1990.

In 1959 he published a counterexample which solved Hilbert's 14th problem and also answered the underlying invariant theoretical question. David Hilbert asked whether the invariant ring of the representation of the general linear group in the polynomial ring in n variables over a field k is finite, which Nagata refuted with a counterexample. In the same essay he formulated the Nagata conjecture for plane algebraic curves. The curve C in two-dimensional projective space should go through r points in a general position with given multiplicities . Then Nagata suspected that for ${\displaystyle k[x_{1},...,x_{n}]}$ ${\displaystyle P_{i}}$ ${\displaystyle m_{i}}$ ${\displaystyle r>9}$

${\displaystyle \deg C>{\frac {1}{\sqrt {r}}}\sum _{i=1}^{r}m_{i}.}$

The assumption is open.

Numerous other counterexamples from commutative algebra and algebraic geometry can be traced back to him. For example, he showed that full algebraic varieties exist in three dimensions that are not projective. In his book Local Rings , a standard work on commutative algebra and algebraic geometry, he introduced the Nagata rings named after him today (he called them pseudogeometric rings at the time), a subclass of Noether's rings .

Works by him in the late 1950s on algebraic geometry on Dedekindrings were important for the later schema theory of the Grothendieck group. His concept of the henselization of rings in the late 1950s and the completion of algebraic varieties (1962) were also influential .

A conjecture by Nagata from 1972 that a certain automorphism of the polynomial ring in 3 variables cannot be composed of certain elementary automorphisms (technically: the Nagata automorphism of the polynomial ring in 3 variables is wild) was proven in 2004 by Ivan Shestakov and Ualbai Umirbaev.

In 1958 he was invited speaker at the International Congress of Mathematicians in Edinburgh ( On the fourteenth problem of Hilbert ). One of his students at Kyoto University was Shigefumi Mori .

He was on the Council of the Japanese Mathematical Society and the Science Council of Japan. From 1979 to 1982 he was Vice President of the International Mathematical Union . In 1961 he received the Chunichi Culture Prize , the Matsunaga Prize in 1970 and the Japanese Academy Prize in 1986. In 1998 he received the Order of the Holy Treasure .

Fonts

• Local Rings , Interscience 1962
• Lectures on the fourteenth problem of Hilbert , Tata Institute of Fundamental Research Lectures on Mathematics 31, Bombay: Tata Institute of Fundamental Research, 1965
• A treatise on the 14th problem of Hilbert , Mem. Coll. Sci. Univ. Kyoto, Ser. A, Volume 30, 1956, pp. 57-70, pp. 197-200 (Addition and Correction)
• On the 14th problem of Hilbert , American Journal of Mathematics, Volume 81, 1959, pp. 766-772

literature

• Masaki Maruyama, Miyanishi Masayoshi, Shigefumi Mori , Tadao Oda: Masayoshi Nagata (1927-2008) , Notices of the American Mathematical Society 56, January 2008, p. 58, pdf

Individual evidence

1. ↑ www.ams.org: Mathematics People Accessed June 16, 2010 (English, PDF; 1.5 MB)
2. ↑ Nagata, On the 14th problem of Hilbert, American Journal of Mathematics 81, 1959, pp. 766-772
3. ↑ Masayoshi Nagata: A generalization of the imbedding problem of an abstract variety in a complete variety. Journal of Mathematics of Kyoto University (3) 1963, 89-102.
4. ↑ Nagata, On automorphism group of , Tokyo 1972 ${\displaystyle k[x,y]}$
5. ↑ Shestakov, Umirbaev, The tame and the wild automorphisms of polynomial rings in three variables, Journal of the American Mathematical Society 17, 2004, pp 197-227