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 Polynomial
Famous quotes containing the words algebraic, forms and/or general:
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)
“The strongest and most effective [force] in guaranteeing the long-term maintenance of ... power is not violence in all the forms deployed by the dominant to control the dominated, but consent in all the forms in which the dominated acquiesce in their own domination.”
—Maurice Godelier (b. 1934)
“We will win this war because we are willing to sacrifice 10,000,000 lives. How many lives is the White Man willing to sacrifice?”
—Jerome Cady, U.S. screenwriter, and Lewis Milestone. General Mitsubi (Richard Loo)