Universal quadratic form
In mathematics, a universal quadratic form is a quadratic form over a ring that represents every element of the ring.Lam (2005) p.10 A non-singular form over a field which represents zero non-trivially is universal.Rajwade (1993) p.146
Examples
- Over the real numbers, the form x2 in one variable is not universal, as it cannot represent negative numbers: the two-variable form {{nowrap|x2 − y2}} over R is universal.
- Lagrange's four-square theorem states that every positive integer is the sum of four squares. Hence the form {{nowrap|x2 + y2 + z2 + t2 − u2}} over Z is universal.
- Over a finite field, any non-singular quadratic form of dimension 2 or more is universal.Lam (2005) p.36
Forms over the rational numbers
The Hasse–Minkowski theorem implies that a form is universal over Q if and only if it is universal over Qp for all p (where we include {{nowrap|1=p = ∞}}, letting Q∞ denote R).Serre (1973) p.43 A form over R is universal if and only if it is not definite; a form over Qp is universal if it has dimension at least 4.Serre (1973) p.37 One can conclude that all indefinite forms of dimension at least 4 over Q are universal.
See also
- The 15 and 290 theorems give conditions for a quadratic form to represent all positive integers.
References
{{reflist}}
- {{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=Graduate Studies in Mathematics | first=Tsit-Yuen | last=Lam | authorlink=Tsit Yuen Lam | publisher=American Mathematical Society | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 | mr = 2104929 }}
- {{cite book | title=Squares | volume=171 | series=London Mathematical Society Lecture Note Series | first=A. R. | last=Rajwade | publisher=Cambridge University Press | year=1993 | isbn=0-521-42668-5 | zbl=0785.11022 }}
- {{cite book | first=Jean-Pierre | last=Serre | authorlink=Jean-Pierre Serre | title=A Course in Arithmetic | series=Graduate Texts in Mathematics | volume=7 | publisher=Springer-Verlag | year=1973 | isbn=0-387-90040-3 | zbl=0256.12001 | url-access=registration | url=https://archive.org/details/courseinarithmet00serr }}