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Thus deg(f⋅g) = 0 which is not greater than the degrees of f and g (which each had degree 1). Since the norm function is not defined for the zero element of the ring, we consider the degree of the polynomial f(x) = 0 to also be undefined so that it follows the rules of a norm in a Euclidean domain.
Hunter College is a public university in New York City, United States. It is one of the constituent colleges of the City University of New York and offers studies in more than one hundred undergraduate and postgraduate fields across five schools. It also administers Hunter College High School and Hunter College Elementary School. [4]
Hunter College in 1874, when it was the Normal College of the City of New York. Founded in 1870 as a teacher's college for women, Hunter College is one of the oldest public higher educational institutions in the United States. More than 23,000 students currently attend Hunter, pursuing undergraduate and graduate degrees in more than 170 areas ...
The restriction in the definition to polynomials of degree greater than one excludes the infinitely many decompositions possible with linear polynomials. Joseph Ritt proved that m = n {\displaystyle m=n} , and the degrees of the components are the same up to linear transformations, but possibly in different order; this is Ritt's polynomial ...
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But the terms of P which contain only the variables X 1, ..., X n − 1 are precisely the terms that survive the operation of setting X n to 0, so their sum equals P(X 1, ..., X n − 1, 0), which is a symmetric polynomial in the variables X 1, ..., X n − 1 that we shall denote by P̃(X 1, ..., X n − 1). By the inductive hypothesis, this ...
Polynomial rings and their quotients by homogeneous ideals are typical graded algebras. Conversely, if S is a graded algebra generated over the field K by n homogeneous elements g 1, ..., g n of degree 1, then the map which sends X i onto g i defines an homomorphism of graded rings from = [, …,] onto S.
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 /. So for example, the following function is positively homogeneous of degree 1 but not homogeneous: ( x 2 + y 2 + z 2 ) 1 2 . {\displaystyle \left(x^{2}+y^{2}+z^{2 ...