Reduce quadratic form of 3 variables to sum of 3 squaresI am dealing with following quadratic form: q:R3→Rq(x,y,z)=xy+yz+xz

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Drahthaare89c · Accepted Answer

As Ben Grossmann said, q is not a sum of squares, but it can always be written as ∑iαiφi2 where αi≠0 and φi is a linear form.Nevertheless, you are in the nasty case (the case where there is no squared terms). The trick is to choose two variables (say x and y), isolate all the terms containing at least one of the two variables , write it under the form A[xy+B(z)x+C(z)y], where A is a nonzero constant, and B,C are terms in z only, then &quot;complete the product&quot;, and use the identity uv=d14((u+v)2−(u−v)2).For you example q(x,y,z)=1.xy+z.x+z.y=(x+z)(y+z)−z2=d14(x+y+2z)2−d14(x−y)2−z2This approach may be generalized to an arbitrary number of variables.

Reduce quadratic form of 3 variables to sum

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