A modular approach to the computation of the number of real roots

The problem of computing the number of distinct real roots of a real polynomial can be solved analyzing the sign variations of the sequence of principal minors of the Hankel matrix associated with the given polynomial. In this paper, we present a modular algorithm to achieve this goal. In this approach, the principal minors sequence of the associated Hankel matrix is computed using modular methods. The computing time analysis shows that the maximum computing time function of the modular algorithm is Ο(n⁵ l²), where n is the degree of the polynomial and l its length.

Let K[x] be a polynomial ring over the unique factorization domain K, and ƒ(x), g(x) ε K[x (deg(ƒ) = n, deg(g) = m, m ≤ n). Then, we mean by the K-Hankel matrix H_n (f, g) associated with {f, g}, the matrix (h_i,j)_1≤i,j≤n, where h_i,j = d_i+j-1 and q(x) = d_n-mx^n+m-1 + … + d_2n-1 is the pseudoquotient of x^2n-1 g(x) and ƒ(x) (d_i = 0 if 1 ≤ i < n - m). We also mean by the Q (K)-Hankel matrix H_n (f, g) associated with {f, g}, the matrix (h_i,j) _1≤i,j≤n, where h_i,j = d_i+j-1 and q(x) = d_n-mx^n+m-1 + … + d_2n-1 is the quotient of x^2n-1g(x) and ƒ(x) in the quotient field Q (K) of K (d_i = 0 if 1 ≤ i < n - m).

If K is the real field and g(x) is the derivative of ƒ(x) (n = deg(ƒ)), then we associate with the real polynomial ƒ the matrix H_n(f, f′). We also denote the i x i principal submatrix of H_n by H_i. We write Δ_i = sgn(D_i), D_i = det(H_i) ≠ 0, and otherwise Δ_i = (-1)^i(i-1)/2 Δ_k, provided that D_k ≠ 0, D_j = 0 for k < j ≤ i.

A classical result due to Frobenious states that the number of distinct real roots of the polynomial ƒ(x) is ν = r - 2V, where V is the number of sign variations of the sequence S = (1, Δ₁, …, Δ_r), and r = rank(H_n). Therefore, an effective method can be achieved computing the sequence of principal minors. This can be done algorithmically in Ο(n²) operations using the fundamental vector sequence.