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For polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree (sometimes called the total degree) of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial x 2 y 2 + 3x 3 + 4y has degree 4, the same degree as the term x ...
Polynomials of small degree have been given specific names. A polynomial of degree zero is a constant polynomial, or simply a constant. Polynomials of degree one, two or three are respectively linear polynomials, quadratic polynomials and cubic polynomials. [8]
For example, if A is a multiple aI n of the identity matrix, then its minimal polynomial is X − a since the kernel of aI n − A = 0 is already the entire space; on the other hand its characteristic polynomial is (X − a) n (the only eigenvalue is a, and the degree of the characteristic polynomial is always equal to the dimension of the space).
The resulting polynomial is not a linear function of the coordinates (its degree can be higher than 1), but it is a linear function of the fitted data values. The determinant, permanent and other immanants of a matrix are homogeneous multilinear polynomials in the elements of the matrix (and also multilinear forms in the rows or columns).
A polynomial in these indeterminates, with coefficients in a field K, or more generally a ring, is a finite linear combination of monomials = with coefficients in K. The degree of a nonzero polynomial is the maximum of the degrees of its monomials with nonzero coefficients.
For example, + + is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions. Given a homogeneous polynomial of degree with real coefficients that takes only positive values, one gets a positively homogeneous function of degree / by raising it to the power /.
In numerical analysis, polynomial interpolation is the interpolation of a given bivariate data set by the polynomial of lowest possible degree that passes through the points of the dataset.
The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any basis (that is, the characteristic polynomial does not depend on the choice of a basis).