# New PDF release: A Bernstein property of solutions to a class of prescribed By McCoy J. A.

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Replace xα with the next monomial in lex order which is not divisible by any of the monomials LT(gi ) for gi ∈ Glex . Exercise 3 below will explain how the Next Monomial procedure works. Now repeat the above process by using the new xα as input to the Main Loop, and continue until the Termination Test tells us to stop. Before we prove the correctness of this algorithm, let’s see how it works in an example. Exercise 1. Consider the ideal I = xy + z − xz, x2 − z, 2x3 − x2 yz − 1 §3. Gr¨ obner Basis Conversion 51 in Q[x, y, z].

00000. Instead of 20 real roots, the new polynomial has 12 real roots and 4 complex conjugate pairs of roots. Note that the imaginary parts are not even especially small! While this example is admittedly pathological, it indicates that we should use care in ﬁnding roots of polynomials whose coeﬃcients are only approximately determined. (The reason for the surprisingly bad behavior of this p is essentially the equal spacing of the roots! ) Along the same lines, even if nothing this spectacularly bad happens, when we take the approximate roots of a one-variable polynomial and try to extend to solutions of a system, the results of a numerical calculation can still be unreliable.

Given the input xα , compute xα . Then: G a. If xα is linearly dependent on the remainders (on division by G) of the monomials in Blex , then we have a linear combination G xα − α(j) j cj x G = 0, where xα(j) ∈ Blex and cj ∈ k. This implies that g = xα − α(j) j cj x ∈ I. We add g to the list Glex as the last element. 3) below), whenever a polynomial g is added to Glex , its leading term is LT(g) = xα with coeﬃcient 1. G b. If xα is linearly independent from the remainders (on division by G) of the monomials in Blex , then we add xα to Blex as the last element.