| Generalized Budan-Fourier theorem and virtual roots (2004) | |||||||||||||
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| In this Note we give a proof of a generalized version of the classical Budan-Fourier theorem, interpreting sign variations in the derivatives in terms of virtual roots. 1 Generalized Budan-Fourier theorem The number of sign changes, V(a), in a sequence, a = a0, · · · , ap, of elements in R \ {0} is defined by induction on p by: V(a0) = 0 ⎨ V(a1, · · · , ap) + 1 if a0a1 < 0 V(a0, · · · , ap) = V(a1, · · · , ap) if a0a1> 0 This definition extends to any finite sequence a of elements in R by considering the finite sequence b obtained by dropping the zeros in a and defining V(a) = V(b), stipulating that V of the empty sequence is 0. | |||||||||||||
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