c Allerton Press, Inc., 2007.
ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2007, Vol. 51, No. 8, pp. 67–70. c V.E. Geit, N.Zh. Geit, 2007, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2007, No. 8, pp. 70–73.
Original Russian Text Nonnegativity Criteria for Fourth Degree Polynomials on the Axis
V. E. Geit1 and N. Zh. Geit2
1
Chelyabinsk State University, ul. Brat’ev Kashirinykh 129, Chelyabinsk, 454021 Russia1
2
Chelyabinsk State Pedagogical University, pr. Lenina 69, Chelyabinsk, 454080 Russia
Received February 7, 2007
DOI: 10.3103/S1066369X07080087
Nonnegative polynomials f (x) on the line R = (−∞, +∞) are characterized by the possibility to be
represented as a sum f (x) = g2 (x) + h2 (x) of squares of two real polynomials g and h (see, e. g., [1],
p. 292). However, up to now, in the general case, no conditions on the coefficients of a polynomial f are
known under which it is a sum of two squares. In this paper, we find such nonnegativity conditions for
fourth degree polynomials on the axis (see Theorems 1–3 and Proposition 1).
1. Initial nonnegativity criteria. First we consider polynomials without the third power of the
variable.
P (x) = x4 + bx2 + cx + d,
b, c, d ∈ R = (−∞, +∞)
(1)
(see Theorem 1). In this case, we can restrict ourselves to the basic case when c = 0 and d > 0 in (1).
In fact, if polynomial (1) is nonnegative everywhere, then d = P (0) ≥ 0. If d = 0, then, for P (x) ≥ 0
on R, we necessarily have c = 0, because otherwise P (x) = x(x3 + bx + c) changes the sign in a
neighborhood of zero. If d = 0, c = 0 in (1), one can easily see that P (x) ≥ 0 everywhere but only for
b ≥ 0. The following theorem holds (see also [2]). Its statement 1) is well-known, we formulate it here
for the sake of completeness.
Theorem 1. Let P (x) = x4 + bx2 + cx + d, where d > 0.
1) for c = 0 and b ≥ 0, we have P (x) > 0 (−∞ < x < +∞), and if c = 0 and b < 0, then P (x) ≥ 0
2
on R if and only if d ≥ b4 ;
2) for c = 0, the following statements hold:
(a) if b ≥ 0, then P (x) ≥ 0 for all real x if and only if
27c2 ≤ 2b(36d − b2 ) + 2(b2 + 12d)3/2 ;
(2)
(b) if b < 0, then in order for P (x) to be nonnegative on the whole axis R, it is necessary and
2
sufficient that relation (2) hold and d > b4 .
Statement 2) can be proved with the use of Lemmas 1–5.
Lemma 1. A polynomial (1) is the square of a polynomial with real coefficients if and only if
2
c = 0 and d = b4 .
Lemma 2 is a partial case of a well-known result (see, e. g., [3], p. 256) on the representation of real
polynomials as a sum of two squares.
1
E-mail: [email protected].
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