Integer Arithmetic Determination of Polynomial Real Roots

f~rmed by an algorith m (detailed below) under which th e degrees no, nl , ... nt of the Pi (x) decrease monotonically. We denote degree drop by d i = ni nj _ 1 ~ 1; we distinguish the lead ing and maximum coefficien t magnitudes of the Pi (x) by Ci = I p\/) I and gi ~ I pI!') I; we deno te the norms of the P;(x) by ej = (~(p\,:»2) 1/2; and we distinguish the degree of the initia l Po(x) by N = no. We observe t ~N, and we defineM=N-l. We consider two algorithms. In the Sturm case: PI (x) is the derivative of Po(x) , and each other P i(x) is the negative of the remainder polynomial upon dividing P i2(X) by P j _ 1 (x); and di ~ 1 yields t ~ N. In the Budan (or Fourier-Budan) case : every Pi(x) for i ~ 1 is the derivative of P j _ I (x); and all d i = 1 yields t = N. We discuss these cases, first together, and then separately. We seek the real roots of Po (x) by determining the pIA) for i ~ 1 and then employing a proced ure which requires determining, for selected values of x, the sign(s) of either Po(x) or all Pj(x). Hence, as convenient , we may replace any Pi (x) by any associate polynomial formed by multiplying Pi(x) by some positive scaling factor Ii. Arbitrarily, we regard PhO) > 0; and (optionally) we ignore zero roots of P o(x) by imposing p~o) ¥ 0 (adjusting N if necessary). Trivially, we demandN > o. By B we denote some bound such that all real roo ts of P o(x) lie in the range B < x ~ B (e.g_, eu < B ); and, by analyzing the signs of the ?i(X) at interval e ndpoints , we develop successively s maller intervals x' < x ~ x" within B < x ~ B to bou nd each root in an interval of desired small len gth. We consider onl y rational p~.?) (e.g., terminating digital expressions), and we restrict all 1; to be rational, when ce all pl/,) are rational.