mathematics

Masayoshi Nagata

Masayoshi Nagata is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Masayoshi Nagata rather than just read about it. In short: Masayoshi Nagata (Japanese: 永田雅宜 Nagata Masayoshi; February 9, 1927 – August 27, 2008) was a Japanese mathematician, known for his work in the field of commutative algebra. He was celebrated for the construction of a number of intricate counterexamples for some seemingly plausible statements in commutative algebra and algebraic geometry, earning him the nickname "Mr.

Key takeaways

Masayoshi Nagata belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Masayoshi Nagata to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Masayoshi Nagata from memory before moving on to harder problems.

Reference excerpt

Masayoshi Nagata (Japanese: 永田雅宜 Nagata Masayoshi; February 9, 1927 – August 27, 2008) was a Japanese mathematician, known for his work in the field of commutative algebra. He was celebrated for the construction of a number of intricate counterexamples for some seemingly plausible statements in commutative algebra and algebraic geometry, earning him the nickname "Mr. Counterexample".

Work Nagata's compactification theorem shows that algebraic varieties can be embedded in complete varieties. The Chevalley–Iwahori–Nagata theorem describes the quotient of a variety by a group. In 1959, he introduced a counterexample to the general case of Hilbert's fourteenth problem on invariant theory. His 1962 book on local rings contains several other counterexamples he found, such as a commutative Noetherian ring that is not catenary, and a commutative Noetherian ring of infinite dimension. Nagata's conjecture on curves concerns the minimum degree of a plane curve specified to have given multiplicities at given points; see also Seshadri constant. Nagata's conjecture on automorphisms concerns the existence of wild automorphisms of polynomial algebras in three variables. Recent work has solved this latter problem in the affirmative.

Selected works Nagata, Masayoshi (1960), "On the fourteenth problem of Hilbert", Proc. Internat. Congress Math. 1958, Cambridge University Press, pp. 459–462, MR 0116056, archived from the original on 2011-07-17 Nagata, Masayoshi (1965), Lectures on the fourteenth problem of Hilbert (PDF), Tata Institute of Fundamental Research Lectures on Mathematics, vol. 31, Bombay: Tata Institute of Fundamental Research, MR 0215828 Nagata, Masayoshi (1962), Local rings, Interscience Tracts in Pure and Applied Mathematics, vol. 13, New York-London: Interscience Publishers a division of John Wiley & Sons, ISBN 0-88275-228-6, MR 0155856 {{citation}}: ISBN / Date incompatibility (help)

References

Maruyama, Masaki; Masayoshi Miyanishi; Shigefumi Mori; Tadao Oda (January 2009). "Masayoshi Nagata (1927–2008)" (PDF). Notices of the American Mathematical Society. 56 (1): 58. Retrieved 2008-12-30. O'Connor, John J.; Robertson, Edmund F., "Masayoshi Nagata", MacTutor History of Mathematics Archive, University of St Andrews Masayoshi Nagata at the Mathematics Genealogy Project

Illustrations

Worked examples

Example 1 — a first encounter with Masayoshi Nagata

Start with the simplest possible case. Write down what Masayoshi Nagata claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Masayoshi Nagata before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Masayoshi Nagata ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Masayoshi Nagata

In research: Masayoshi Nagata appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Masayoshi Nagata in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Masayoshi Nagata is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1927 births, 2008 deaths, 20th-century Japanese mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Masayoshi Nagata outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Masayoshi Nagata in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Masayoshi Nagata means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Masayoshi Nagata out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Masayoshi Nagata in simple terms?

Why does Masayoshi Nagata matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Masayoshi Nagata?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Masayoshi Nagata.