La luz es polar : Projective geometry and real polynomial equation solving ?

The main outcome of this paper is the following: Let Q [X1, . . . , Xn] be the ring of n –variate polynomials over the rational numbers Q and let F1, . . . , Fp with 1 ≤ p ≤ n be given polynomials of Q [X1, . . . , Xn] of degree at most d . Suppose that F1, . . . , Fp are represented by a division–free arithmetic circuit of size L and non–scalar depth ` over Q . Furthermore, assume that the polynomials F1, . . . , Fp form a regular sequence in Q [X1, . . . , Xn] , that, for each 1 ≤ h ≤ p , the ideal generated by F1, . . . , Fh is radical and that the real algebraic variety SR defined by F1, . . . , Fp in A := R is non–empty and smooth. Then there exists an arithmetic network N with ” = ” and ” < ” decision gates over Q , which finds a (suitably encoded) representative point for each connected component of SR . The size and non–scalar depth of N are bounded by ( n p ) L(ndδ) and O(n(`+log nd) log δ) , respectively, where δ ≤ d pn−p is the (suitably defined) degree of the real interpretation of the polynomial equation system F1 = · · · = Fp = 0 . In order to prove this result we introduce the new notion of generalized, dual and conic polar varieties of equidimensional closed algebraic subvarieties of the real and complex affine and projective spaces.