The check symbols are coefficients of the

monic polynomial (the original data is sliced to the roots of the polynomial).

[c.sub.m](z) be arbitrary

monic polynomials over R[z] which satisfy the conditions:

The reflection vectors of a Schur stable

monic polynomial a(z) are defined as the end points of stable line segments [A.sup.i]([+ or -] 1) = conv{a|[k.sub.i] = [+ or -] 1} [9]

The reflection vectors of a Schur stable

monic polynomial f(z) are defined as the end points of the stable line segments conv{f|[k.sub.i](f) = [+ or -]1}

Consider the

monic polynomial f(x) = (x - 3)(x - 2).

The type II multiple orthogonal polynomial with respect to (W, n) is the

monic polynomial [P.sub.n](x) of degree |n| that satisfies the following orthogonality conditions:

where [F.sub.k,D](x) is a

monic polynomial in x of degree D = j + m with integer coefficients.

Section 2 describes our first interpretation for the cyclotomic polynomial, which applies much more generally to any

monic polynomial in Z[x].

The (type II) multiple orthogonal polynomial [P.sub.[??]] is the

monic polynomial of degree |[??]| = [n.sub.1] + [n.sub.2] + * * * + [n.sub.r] that satisfies the orthogonality conditions

There exist [PHI], a nonzero

monic polynomial of degree t, [PSI], a polynomial of degree p, and B, a polynomial of degree r, such that

Theorem 2.1 If 0 [less than or equal to] n [less than or equal to] 22, then for every

monic polynomial P [member of] Z[x] of degree n there exists an irreducible

monic polynomial Q [member of] Z[x] of degree n such that |P - Q| [less than or equal to] 4.

It is based on the companion matrix of a

monic polynomial over H,

Notice that [[DELTA].sub.m] is a divisor of the discriminant of the polynomial [[LAMBDA].sup.3] + 4[[[LAMBDA].sup.2] - (12m - 4)[LAMBDA] + 4[m.sup.2] (obtained by rewriting the equation P([LAMBDA], m) = 0 as a

monic polynomial equation in 2/[LAMBDA]) and this last discriminant is -16(27[m.sup.4] - 216[m.sup.3] + 280[m.sup.2] - 48m).