Algebraic Forms in General
Algebraic form, or simply form, is another term for homogeneous polynomial. These then generalise from quadratic forms to degrees 3 and more, and have in the past also been known as quantics (a term that originated with Cayley). To specify a type of form, one has to give the degree d and the number of variables n. A form is over some given field K, if it maps from Kn to K, where n is the number of variables of the form.
A form f over some field K in n variables represents 0 if there exists an element
- (x1,...,xn)
in Kn such that f(x1,...,xn) =0 and at least one of the xi is not equal to zero.
A quadratic form over the field of the real numbers represents 0 if and only if it is not definite.
Read more about this topic: Homogeneous Polynomials
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