NEW BOUNDS ON THE REAL POLYNOMIAL ROOTS

The presented analysis determines several new bounds on the real roots of the equation a_nxⁿ + a_n−1xⁿ⁻¹ + · · · + a₀ = 0 (with a_n > 0). All proposed new bounds are lower than the Cauchy bound max ⁿ 1^P_jⁿ₌₀⁻¹ |a_j/a_n| ^o . Firstly, the Cauchy bound formula is derived by presenting it in a new light – through a recursion. It is shown that this recursion could be exited at earlier stages and, the earlier the recursion is terminated, the lower the resulting root bound will be. Following a separate analysis, it is further demonstrated that a significantly lower root bound can be found if the summation in the Cauchy bound formula is made not over each one of the coefficients a₀, a₁, . . ., a_n−1, but only over the negative ones. The sharpest root bound in this line of analysis is shown to be the larger of 1 and the sum of the absolute values of all negative coefficients of the equation divided by the largest positive coefficient. The following bounds are also found in this paper: max { 1, (^Pq_j=1 B_j/A_l )¹/(^l−k)} , where B₁, B₂, . . ., B_q are the absolute values of all of the negative coefficients in the equation, k is the highest degree of a monomial with a negative coefficient, A_l is the positive coefficient of the term A_lx^l for which k < l ≤ n.