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Nagata's Conjecture
In algebra, Nagata's conjecture states that Nagata's automorphism of the polynomial ring ''k'' 'x'',''y'',''z''is wild. The conjecture was proposed by and proved by . Nagata's automorphism is given by :\phi(x,y,z) = (x-2\Delta y-\Delta^2z,y+\Delta z,z), where \Delta=xz+y^2. For the inverse, let (a,b,c)=\phi(x,y,z) Then z=c and \Delta= b^2+ac. With this y=b-\Delta c and x=a+2\Delta y+\Delta^2 z. References * *{{Citation , last1=Umirbaev , first1=Ualbai U. , last2=Shestakov , first2=Ivan P. , title=The tame and the wild automorphisms of polynomial rings in three variables , doi=10.1090/S0894-0347-03-00440-5 , mr=2015334 , year=2004 , journal=Journal of the American Mathematical Society The ''Journal of the American Mathematical Society'' (''JAMS''), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society. It was established in January 1988. Abstracting and indexing This journal is abstr ... , issn=0894-0347 , volume=17 , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Masayoshi Nagata
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. Work Nagata's compactification theorem shows that 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 so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variable (mathematics), variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in mathematical education, education, to the study of algebraic structures such as group (mathematics), groups, ring (mathematics), rings, and field (mathematics), fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wild Automorphism
Wild, wild, wilds or wild may refer to: Common meanings * Wild animal * Wilderness, a wild natural environment * Wildness, the quality of being wild or untamed Art, media and entertainment Film and television * ''Wild'' (2014 film), a 2014 American film from the 2012 book * ''Wild'' (2016 film), a 2016 German film * '' The Wild'', a 2006 Disney 3D animation film * ''Wild'' (TV series), a 2006 American documentary television series * The Wilds (TV series), a 2020 fictional television series Literature * '' Wild: From Lost to Found on the Pacific Crest Trail'' a 2012 non-fiction book by Cheryl Strayed * ''Wild, An elemental Journey'', a 2006 autobiographical book by Jay Griffiths * ''The Wild'' (novel), a 1991 novel by Whitley Strieber * ''The Wild'', a science fiction novel by David Zindell * ''The Wilds'', a 1998 limited-edition horror novel by Richard Laymon Music * ''Wild'' (band), a five-piece classical female group Albums and EPs * ''Wild'' (EP), 2015 * ''Wild'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The American Mathematical Society
The ''Journal of the American Mathematical Society'' (''JAMS''), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society. It was established in January 1988. Abstracting and indexing This journal is abstracted and indexed in: 2011. American Mathematical Society. * * * * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other result ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
