On the Piano Movers Problem Ii General Techniques for Computing Topological P
On the Piano Movers Problem Ii General Techniques for Computing Topological P
Jacob T Schwartz
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-51- The technique for resultant calculation given in [Sc] depends only on the identities (l)-(5) and hence can be adapted to the calculation of subresultants. Fleshing out this summary remark, we shall now present an efficient technique for the simultaneous calculation of all subresultants of a given pair of polynomials P and Q. To this end, we make the following definition. Definition 2l ^^^ a pair of polynomials w = [P, Q] of degree d, d' with coefficients in a field F be given, and let d = ...max(d, d'). Write P mod Q for the remainder of P upon division by Q. Then the RQ-sequence RQ(w) of w is the sequence tj, i = d, d-l, ... , 0, of quadruples ti = [[Pi. Qi], a^, bi, M^], defined as follows: (1) P^, Q^ are polynomials, a^ is a quantity of F, b^^ is always +1 or -1, and >L is a 2x2 matrix of polynomials with coefficients in F. (2) tj = [ [P, Q], 1, 1, 1], where I is the 2 x 2 identity matrix. (3) roax(deg(P^), deg(Q^)) > i > min(deg(P^), deg(Q^) ) for i > 0. (4a) If min(deg(Pi), deg(Qi))
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