Thue equation

In mathematics, a Thue equation is a Diophantine equation of the form

ƒ(x,y) = r,

where ƒ is an irreducible bivariate form of degree at least 3 over the rational numbers, and r is a nonzero rational number. It is named after Axel Thue who in 1909 proved a theorem, now called Thue's theorem, that a Thue equation has finitely many solutions in integers x and y.[1]

The Thue equation is solvable effectively: there is an explicit bound on the solutions x, y of the form where constants C1 and C2 depend only on the form ƒ. A stronger result holds, that if K is the field generated by the roots of ƒ then the equation has only finitely many solutions with x and y integers of K and again these may be effectively determined.[2]

Solving Thue equations

Solving a Thue equation can be described as an algorithm[3] ready for implementation in software. In particular, it is implemented in the following computer algebra systems:

  • in PARI/GP as functions thueinit() and thue().
  • in Magma computer algebra system as functions ThueObject() and ThueSolve().
  • in Mathematica through Reduce
gollark: I think it is written somewhere that anything you promise to do is considered, well, binding by the eldræ, so that's not massively far off.
gollark: We should just get rid of the non-cubicley toilets.
gollark: (probably not, but it would be kind of ironic)
gollark: Random idea: maybe people's belief in the bystander effect *causes* the bystander effect.
gollark: Unless the universe is just being simulated by accident as part of solving some complex optimization problem or something weird like that.

See also

References

  1. A. Thue (1909). "Über Annäherungswerte algebraischer Zahlen". Journal für die reine und angewandte Mathematik. 1909 (135): 284–305. doi:10.1515/crll.1909.135.284.
  2. Baker, Alan (1975). Transcendental Number Theory. Cambridge University Press. p. 38. ISBN 0-521-20461-5.
  3. N. Tzanakis and B. M. M. de Weger (1989). "On the practical solution of the Thue equation". Journal of Number Theory. 31 (2): 99–132. doi:10.1016/0022-314X(89)90014-0.

Further reading


This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.