Monic polynomial

Recalling that the companion matrix of any monic polynomial, e x)... 

PROOF. Let us denote by Z the smallest unitary subring of C. The C-module CS is finitely generated. Thus, as is zero of a monic polynomial over Z. Thus, each characteristic root of as is integral over Z. Thus, as x(crs) s a sum °f characteristic roots of as, the claim follows from Theorem 8.2.4. 

The alternative recursion (57) involves the monic Lanczos states ] ) Here, the term monic" serves to indicate that for any given integer n, the highest state ) in the finite sum, which defines the vector jr ), always has an overall multiplying coefficient equal to unity similarly to a monic polynomial [2], The Lanczos states ) are orthogonal, but unnormalized, as opposed to orthonormalized Lanczos states ( linear combinations of powers of the operator U(r) acting on the initial state 0)- Therefore, due to the relation 4> ) = U (r) 0) from Eq. (36), the vectors tjrn) and [ ) are certain sums of Schrodinger states ). ... 

A sequence is monotone if its terms are increasing or decreasing, monic polynomial... 

The recursive relation is the most important property for constructive and computational use of orthogonal polynomials. In fact, as will be shown below, knowledge of the recursion coefficients allows the zeros of the orthogonal polynomials to be computed, and with them the quadrature rule. Therefore the calculation of the coefficients of this three-term recurrence relation is of paramount importance. The recursive relationship in Eq. (3.5) generates a sequence of monic polynomials that are orthogonal with respect to the weight function... 

If, in particular, we convert a matrix L —xl into its SCF, where now the (0,1)-entries of L are elements of -Fig], and factor the similarity invariants into products of powers of monic irreducible polynomials Pj x), so that fi x) =... 

The integration domain and the weight function n( ) uniquely define the family of polynomials PaiO)- A polynomial is defined as monic when its leading coefficient (i.e. kap) is equal to unity. Below two important theorems (without proof) are reported. 

Example B. Spec (fc[X]) the affine line over k. This is denoted A. k[X] has 2 types of prime ideals (o) and (f(X)), / an an irreducible polynomial. Therefore Spec (A [X]) has one closed point for each monic irreducible polynomial, and one generic point [(o)] whose closure is all of Spec (fc[X]). Assume k is algebraically closed. Then the closed points are all of the form [(X — o)] we call this the point X = a , and we find that A is just the ordinary X-line together with a generic point. The most general proper closed set is just a finite union of closed points. 

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