Read e-book online Algorithms in Real Algebraic Geometry PDF

By Saugata Basu

ISBN-10: 3662053551

ISBN-13: 9783662053553

ISBN-10: 3662053578

ISBN-13: 9783662053577

This is the 1st graduate textbook at the algorithmic facets of genuine algebraic geometry. the most principles and methods provided shape a coherent and wealthy physique of information. Mathematicians will locate appropriate information regarding the algorithmic facets. Researchers in machine technology and engineering will locate the necessary mathematical history. Being self-contained the booklet is available to graduate scholars or even, for necessary components of it, to undergraduate scholars. This moment version comprises a number of fresh effects on discriminants of symmetric matrices and different proper topics.

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Example text

Thus Q(Xl, ... , Xk) = R(eI, ... , ek) E K. 14: (i) =} (ii) Let P E R[X] of degree p = 2m n with n odd. We show by induction on m that P has a root in R[i]. If m = 0, then p is odd and P has a root in R. Suppose the result is true for m - 1. Let Xl, ... , X p be the roots of P (counted with multiplicities) in an algebraically closed field containing R. For every h E Z, let Qh(XI , ... ,Xp,X) = II (X - X>.. <1-' The coefficients of the polynomial Q h(X I, ... , X p, X) are symmetrie in XI, ... 20 Qh(XI,""Xp,X) E R[X].

The Sturm-query of Q for P in (a, b) is the number SQ(Q,Pja,b) = #({x E (a,b) I P(x) = O/\Q(x) > O})#({XE (a,b) I P(x) =O/\Q(x)

2 The Cauchy Index Let P be a non-zero polynomial with coefficients in areal closed field R. Not only would we like to determine whether P has a root in R but also to determine whether P has a root at which another polynomial Q is positive. With this goal in mind, it is profitable to look at the jumps (discontinuities) of the rational function P(c) = 0, Q(c) I- o. P~Q = iQ~cl + Re, If P~Q . L then where Re is a rational function defined at c. It is now 44 2 Real Closed Fields obvious that if Q(c) 0, then P'Q p jumps from -00 to +00 at c, and if pP'Q jumps from +00 to -00 at c.

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Algorithms in Real Algebraic Geometry by Saugata Basu


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