Extension of the Fourier-Budan theorem to one-variable signomials

Let f(x) = a0x r0 + a1x r1 + · · · + akx rk , where each ai ∈ R, each rk ∈ N := {0, 1, . . . }, and r0 < r1 · · · < rk. Suppose u < v. Let z(f, u, v) = the number of roots of f in (u, v], counted with multiplicity. For any w ∈ R and n ∈ N, let s(f, w, n) = the number of sign-changes in the sequence f(w), f ′(w), f ′′(w), . . . , f (w) (skipping over zeros). Then the Fourier-Budan Theorem says that z(f, u, v) ≤ s(f, u, rk) − s(f, v, rk) and z(f, u, v) ≡ s(f, u, rk) − s(f, v, rk) (mod 2). In this paper we weaken the hypothesis of this theorem by allowing the exponents of f to be arbitrary real numbers; but we must then restrict u and v to be positive, to avoid non-real values of f(x). Our conclusion is then that there exists an N ∈ N such that for all n ≥ N , z(f, u, v) ≤ s(f, u, n) − s(f, v, n) and z(f, u, v) ≡ s(f, u, n) − s(f, v, n) (mod 2).