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
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 very existence of society depends on the fact that every member of it tacitly admits he is not the exclusive possessor of himself, and that he admits the claim of the polity of which he forms a part, to act, to some extent, as his master.”
—Thomas Henry Huxley (1825–95)
“We raised a simple prayer
Before we left the spot,
That in the general mowing
That place might be forgot;
Or if not all so favored,
Obtain such grace of hours
That none should mow the grass there
While so confused with flowers.”
—Robert Frost (1874–1963)