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In this paper we obtain several upper bounds of the moduli of the roots of the algebraic equations. The bounds of the moduli of the roots of the algebraic equations were investigated by many authors (see e.g. [1], [2], [3], [4]). In this paper, for the algebraic equation (1) z + a1z + a2z + · · · + an−1z + an = 0, the following theorems are proved. In the next let (2) Ak = |ak|, k = 1, 2, . . . , n. Theorem 1. Let (3) M1 = max(Ak − Ak−1), k = 2, 3, . . . , n. Then the upper bound for the moduli of the roots of the equation (1) is (B1) 1 + A1, if M1 ≤ 0, (B2) 2 + A1 + √ A 2 1 + 4M1 2 , if M1 > 0. Theorem 2. Let (4) M2 = max(Ak − A1), k = 2, 3, . . . , n and (5) M3 = max ( Ak − A1 k − 1 ) , k = 2, 3, . . . , n. Then the upper bound for the moduli of the roots of the equation (1) is (B3) 1 + A1, if M2 ≤ 0, (B4) 1 + A1 + √ (1 + A1) + 4M2 2 , if M2 > 0, 2000 Mathematics Subject Classification. Primary: 12D10.

A remark on the moduli of the roots of algebraic equations