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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.73 Quadratic Forms

Ben Green

A quadratic form is a homogeneous polynomial of degree 2 in some finite set of unknowns xl, x2,..., xn: an example is q(xl, x2, x3) = Image - 3x1x2 + 4Image. Here, the coefficients 1, -3, and 4 are integers, but the idea generalizes straightforwardly from Image to any ring R. Since linear functions are undeniably important and 2 is the next positive integer after 1, one might expect quadratic forms to be important as well, and indeed ...

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