By Michel Herve
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Extra info for Analytic and plurisubharmonic functions in finite and infinite dimensional spaces. Course given at the University of Maryland, Spring 1970
N . Prove that an−1 = −(α1 + α2 + · · · + αn ) and a0 = (−1)n α1 · α2 · · · αn . A. A theoretical investigation of a polynomial. When n = 2 the formulæ are known from high school . D. Write P (x) in two ways and identify the coeﬃcients. I. By a simple combinatoric argument we get P (x) = (x − α1 )(x − α2 ) · (x − αn ) = x2 − (α1 + · · · + αn )xn−1 + · · · + (−1)n α1 · · · αn = x2 + an−1 xn−1 + · · · + a1 x + a0 . e. the sum of the roots is always equal to −an−1 , and the product of the roots is always equal to (−1)n a0 .
There exists a polynomial Q(x), such that P (x) = (x − α)n Q(x), Q(α) = 0. com 56 Calculus 1c-3 Decomposition Then by successive diﬀerentiations, P (x) = n(x − α)n−1 Q(x) + (x − α)n Q1 (x), P (x) = n(n − 1)(x − α)n−2 Q(x) + (x − α)n−1 Q2 (x), .. P (n−1) (x) P (n) (x) = n(n − 1) · · · 2 · (x − α)Q(x) + (x − α)2 Qn−1 (x), = n! Q(x) + (x − α)Qn (x), where Q1 (x), · · · , Qn (x) are polynomials occurring by the diﬀerentiations. Due to the factor x − α it follows that P (α) = P (α) = P (α) = · · · = P (n−1) (α) = 0, and P (n) (α) = n!
Q(x) x − x2 − 2x Write a MAPLE programme which gives the decomposition. A. Splitting of polynomials followed by a decomposition. D. Guess the roots in (1) and (2). In the latter fraction we factorize the denominator and then decompose. I. 1) It is immediately seen by inspection that the roots are x = 0 and x = 1, hence x(x − 1) = x 2 − x is a divisor in the polynomial: x4 − 3x3 − 6x2 + 8x = x(x − 1)(x2 − 2x − 8). Since x2 − 2x − 8 = (x − 4)(x + 2), we get the factorization x4 − 3x3 − 6x2 + 8x = (x + 2)x(x − 1)(x − 4), and the four simple roots are −2, 0, 1 and 4.
Analytic and plurisubharmonic functions in finite and infinite dimensional spaces. Course given at the University of Maryland, Spring 1970 by Michel Herve