Kronecker's smart, little black boxes
M Giusti, Joos Heintz · Foundations of Computational Mathematics · 2001
This paper is devoted to the complexity analysis of certain uniformity properties owned by all known symbolic methods of parametric polynomial equation solving (geometric elimination). It is shown that any parametric elimination procedure which is parsimonious with respect to branchings and divisions must necessarily have a non-polynomial sequential time complexity, even if highly efficient data structures (as e.g. the arithmetic circuit encoding of polynomials) are used. Introduction Origins, development and interaction of modern algebraic geometry and commutative algebra may be considered as one of the most illustrative examples of historical dialectics in mathematics. Still today, and more than ever before, timeless idealism (in form of modern commutative algebra) is bravely struggling whith secular materialism (in form of complexity issues in computational algebraic geometry). Kronecker was doubtless the creator of this eternal battle field and its first war lord. In a similar way as Gauss did for computational number theory, Kronecker laid intuitively the mathematical foundations of modern computer algebra. He introduced 1882 in [30] his famous “elimination method” for polynomial equation systems and his “parametric representation” of (equidimensional) algebraic varieties. By the way, this parametric representation was until 10 years ago rediscovered again and again. It entered in modern computer algebra as “Shape Lemma” (see e.g. [38, 8, 14, 27]). Using his elimination method in a highly skillful, but unfortunately inimitable way, Kronecker was able to state and to prove a series of fundamental results on arbitrary algebraic varieties.