Abstract
Let p, q be two polynomials with complex coefficients and deg q ≤ deg p = m. This assumption will remain valid through all of this section. Consider the equation
where u, v are polynomials with deg u, deg v ≤ m - 1. If p and q are coprime, the equation has only the trivial solution u = v = 0. If not, denote d = gcd(p, q), deg d = k ≥ 1 and write p = v 0 d , q = - u 0 d; the degree of v 0 equals m - k. The polynomials u 0, v 0 are coprime and satisfy (9.1): u 0 p + v 0 q = 0. If (u,v) is any solution of (9.1), then uv 0 - vu 0, and (u, v) = (ru 0, rv 0) with some polynomial factor r, deg r≤k - 1.
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© 1995 Springer Science+Business Media Dordrecht
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Livšic, M.S., Kravitsky, N., Markus, A.S., Vinnikov, V. (1995). Bezoutian Vessels in Banach Space. In: Theory of Commuting Nonselfadjoint Operators. Mathematics and Its Applications, vol 332. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8561-3_9
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DOI: https://doi.org/10.1007/978-94-015-8561-3_9
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