Almost zeros of a set of polynomials of R[s]

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Abstract
The notion of an exact zero of a sot of polynomials p of R[s] is extended to that of an almost zero. The almost zeros are defined as minima of a norm function of p and their invariance under a. certain equivalence relation &, defined on p, is established, The region of the complex plane which contains the prime almost zero (global minimum) is defined, and necessary and sufficient conditions characterizing the set of almost zeros are given. The role of almost zeros of p in the zero assignment problem of the combinants of p is investigated. Under the assumption of non-assignability for p, it is shown that the almost zeros act as nearly fixed zeros of the combinants. Every almost zero z is the centre of a disc D(z, R) which contains at least one zero of all combinants of p. The size of the radius R of this disc provides a measure of the ability of the almost zero to attract the zeros of the combinants of p A number of results for estimating upper bounds for the radius R, associated with subfamilies of the general family of combinants of p, are given. Finally, a sufficient condition for estimating an upper bound for the disc D(z, R) which contains at least one zero of all combinants of P is derived. The paper provides a more meaningful definition, from the numerical point of view, for the non-generic notion of a zero of a set of polynomials ; it also shows how some of the properties of a zero extend naturally to the almost zero case.

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