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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 ...
In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree n if all of its non-zero terms have degree n. The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined. [c] For example, x 3 y 2 + 7x 2 y 3 − 3x 5 is homogeneous of degree 5. For more details, see ...
There are n Hurwitz determinants for a characteristic polynomial of degree n. See also ... H. S. (1945), "Polynomials whose zeros have negative real parts", ...
Let () be a polynomial equation, where P is a univariate polynomial of degree n. If one divides all coefficients of P by its leading coefficient c n , {\displaystyle c_{n},} one obtains a new polynomial equation that has the same solutions and consists to equate to zero a monic polynomial.
For any given n ≥ 1, among the polynomials of degree n with leading coefficient 1 (monic polynomials): = is the one of which the maximal absolute value on the interval [−1, 1] is minimal. This maximal absolute value is: 1 2 n − 1 {\displaystyle {\frac {1}{2^{n-1}}}} and | f ( x ) | reaches this maximum exactly n + 1 times at: x = cos ...
In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of and is not a divisor of for any k < n. Its roots are all n th primitive roots of unity e 2 i π k n {\displaystyle e^{2i\pi {\frac {k}{n}}}} , where k runs over the positive integers less ...
Furthermore, if the polynomial has a degree 2d greater than two, there are significantly many more non-negative polynomials that cannot be expressed as sums of squares. [4] The following table summarizes in which cases every non-negative homogeneous polynomial (or a polynomial of even degree) can be represented as a sum of squares:
In mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field is a linear combination of positive and negative powers of the variable with coefficients in . Laurent polynomials in X {\displaystyle X} form a ring denoted F [ X , X − 1 ] {\displaystyle \mathbb {F} [X,X^{-1}]} . [ 1 ]