algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

, Nagata's conjecture states that Nagata's automorphism of the

polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...

''k'' 'x'',''y'',''z''is

wild Wild, wild, wilds or wild may refer to: Common meanings * Wilderness, a wild natural environment * Wildlife, an undomesticated organism * Wildness, the quality of being wild or untamed Art, media and entertainment Film and television * ''Wild ...

. The

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

was proposed by and proved by . Nagata's

automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all 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 Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse, the inverse of a number that, when added to the ...

, 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 abs ...

, issn=0894-0347 , volume=17 , issue=1 , pages=197–227, doi-access=free Field (mathematics) Theorems in algebra Conjectures that have been proved