Meyer's theorem

In number theory, Meyer's theorem on quadratic forms states that an indefinite quadratic form Q in five or more variables over the field of rational numbers nontrivially represents zero. In other words, if the equation

Q(x) = 0
For the theorem in computational complexity theory, see P/poly.

has a non-zero real solution, then it has a non-zero rational solution (the converse is obvious). By clearing the denominators, an integral solution x may also be found.

Meyer's theorem is usually deduced from the Hasse–Minkowski theorem (which was proved later) and the following statement:

A rational quadratic form in five or more variables represents zero over the field Qp of the p-adic numbers for all p.

Meyer's theorem is best possible with respect to the number of variables: there are indefinite rational quadratic forms Q in four variables which do not represent zero. One family of examples is given by

Q(x1,x2,x3,x4) = x12 + x22 − p(x32 + x42),

where p is a prime number that is congruent to 3 modulo 4. This can be proved by the method of infinite descent using the fact that if the sum of two perfect squares is divisible by such a p then each summand is divisible by p.

See also

References
  • Meyer, A. (1884). "Mathematische Mittheilungen". Vierteljahrschrift der Naturforschenden Gesellschaft in Zürich. 29: 209–222.
  • Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016.
  • Serre, Jean-Pierre (1973). A Course in Arithmetic. Graduate Texts in Mathematics. 7. Springer-Verlag. ISBN 0-387-90040-3. Zbl 0256.12001.
  • Cassels, J.W.S. (1978). Rational Quadratic Forms. London Mathematical Society Monographs. 13. Academic Press. ISBN 0-12-163260-1. Zbl 0395.10029.
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