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By Foata D., Han G.-N.

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Additional info for A basis for the right quantum algebra and the “1 = q” principle

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Since PI is monic with respect to X k , its leading coefficient in Xk never vanishes, and so the annihilation of the resultant implies that for every a E K the polynomials PI(X',Xk) and Q(a,x',X k ) have a common root in K. Since PI(X',Xk ) has a finite number of roots in K, one of them, say 0:, is a root of Q(a, x', X k ) for infinitely many a E K. Choosing s - 1 such distinct elements al, ... , as~ I, we get that the polynomial Q(o:, U) of degree ~ s - 2 in U has s - 1 distinct roots, which is possible only ifQ(x',o:,U) is identically zero.

The univariate polynomial L:~:ül aiti = has n distinct roots and is of degree at most n - 1, hence is the zero polynomial. Thus 1, u, ... ,un - l are linearly independent elements of A and n is at most the dimension of Aas a K-vector space. 1. 2 First Recipes for Polynomial System Solving This section is devoted to showing how to use Gröbner bases to solve polynomial systems. Given a reduction ordering (cf. Chapter 1) < on the monoid of all monomials of K[X], it is possible to define the division of a polynomial Q E K[X] by a finite family of polynomials P.

14), which teIls us that Y = «1. The next equation up (namely b3 = 0) teIls us X = Y. This completely determines the solution set: 22 A. M. 9. Consider b = (X 2 + y 2 + Z2,XYZ) E K[X,Y,Zj2, where K is an effective subfield of C. The reduced Gröbner basis with respect to the lexicographic arder with X > Y > Z can be easily computed by use of a computer algebra software package. The result is B = {bI, b2 , bd where b3 = y 3 Z + Z 3 Y. Since B has nO univariate polynomial in Z, we have Z (B n K[Z]) = Z(0) = K 3 .

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