The Shub-Smale Tau Conjecture is a hypothesis relating the number of integral roots of a polynomial f in one variable and the Straight-Line Program (SLP) complexity of f. A consequence of the truth of this conjecture is that, for the Blum-Shub-Smale model over the complex numbers, P differs from NP. We prove two weak versions of the Tau Conjecture and in so doing show that the Tau Conjecture follows from an even more plausible hypothesis. Our results follow from a new p-adic analogue of earlier work relating real algebraic geometry to additive complexity. For instance, we can show that a nonzero univariate polynomial of additive complexity s can have no more than 15+s^3(s+1)(7.5)^s s! =O(e^ s s ) roots in the 2-adic rational numbers Q_2, thus dramatically improving an earlier result of the author. This immediately implies the same bound on the number of ordinary rational roots, whereas the best previous upper bound via earlier techniques from real algebraic geometry was a quantity in Omega((22.6)^ s^2 ). This paper presents another step in the author's program of establishing an algorithmic arithmetic version of fewnomial theory.
Let Or(n) (the cost of n) be the minimum number of arithmetic operations needed to obtain n starting from 1. We prove that Or(n) > log n -log log n for almost all n E N, and, given ? > 0, Or(n) 1 there are i, j, 0 0, we have T((n) > (loglo gn)l+e for almost all n C N . Here we improve the above results, by proving the following: For almost all n c N we have Tr(n) > lolg nand, for any given E > 0, we have T((n) 0, T(n) > logn + (1 -)log nloglog logn for almost all n c N, log log n (log log n) 2 T((n) < log n + (3 + s log n log log log n for n large enough. log log n (log log n) 2 foiag nuh Moreover, the first inequality does not depend on the number of binary operations that we can use (provided that this number is finite). Received by the editors October 25, 1994 and, in revised form, August 15, 1995. 1991 Mathematics Subject Classification. Primary 11Y16; Secondary 68Q25, 68Q15, 11B75. ( 1997 American Mathematical Society A simplified rotor design utilizing two or less cavities per sample analysis station is described. Sample or reagent liquids are statically loaded directly into respective sample analysis cuvettes by means of respective apertures and centripet al ramps communicating with each cuvette. According to one embodiment, a single static loading cavity communicates with each sample analysis cuvette in a conventional manner to facilitate dynamic transfer of liquid from that cavity to the cuvette where mixing of sample and reagent liquids and their photometric analysis take place. Dynamic loading of sample or reagent liquids is provided in another embodiment.