Horner’s Rule Is Optimal for Polynomial Nullity

The value VF,n(a0, . . . , an, b) = a0 + a1b+ a2b 2 + · · ·+ anb of a polynomial of degree n ≥ 1 over a field F can be computed by Horner’s rule using no more than nmultiplications in F , and it is optimal (for many fields, including the reals R and the complexes C) for the number of multiplications and/or divisions required, cf. Pan [1966], Winograd [1967, 1970]. We show that Horner’s rule is similarly optimal for polynomial nullity, i.e., the decision problem for the relation NF,n(a0, . . . , an, b) ⇐⇒ a0 + a1b+ a2b + · · ·+ anb = 0. The result holds for all usual computation models and (arguably) for all algorithms which decide NF,n(~a, b) for all ~a, b from the field primitives 0, 1,+,−, ·,÷ and =. §