Premium
Determination of a binary quadratic form by its values at integer points
Author(s) -
Watson G. L.
Publication year - 1979
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300009621
Subject(s) - mathematics , integer (computer science) , combinatorics , converse , binary quadratic form , quadratic equation , binary number , quadratic form (statistics) , quadratic function , arithmetic , geometry , computer science , programming language
Let f = f ( x, y ) be a quadratic form with real coefficients in two integer variables x , y . Let V ( f ) be the set of values taken by f ( x, y ) at points ( x, y ) ≠ (0,0). Impose the same conditions on a second form f ′. Trivially, f equivalent to f ′ implies V ( f ) = V ( f ′). It will be shown that the converse implication holds in general for definite forms; the obvious exception f = x 2 + xy + y 2 , f′ = x 2 + 3 y 2 will be shown to be essentially the only one.