N Polynomials with Positive
نویسنده
چکیده
All polynomials in this paper are supposed to have real coefficients. Polynomials which can be represented in the form P(X) = c ad1-x) " U + XY, with all akl > 0 or all akl < 0, (1) k+l 0 have been introduced and studied by 6.6. Lorentz i[ I]; we shall call them polynomials with positive or negative (more exactly non-negative or non-positive) coefficients, respectively, or simply Lorentz polynomials. Their set will be denoted by L + and L-, respectively; also let L = k + u L-. The representation (1) is not unique, since multiplying by we obtain other representations. Among all of the representations (I) of a fixed polynomial p(x) EL, consider those for which m is the least value. This will be called the Lorentz degree of p(x), and it will be by d(p). If ZI " denotes the set of polynomials of degree at most n, then obviously, p(x) E IZ,\ 17, _, implies d(p) 3 Iz. The representation (1) of a p(x) E L with m = d(p) is still not unique, since terms in (1) with k + I< m can be multiplied by (2) with s=m-k-131, resulting in a new representation. method each representation can be transformed into d(p) p(x)= 1 a,(l-x)k(l+x)d(P)-k, all ak > 0 or all ak < 8, (41 k=O 107
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