# Upperbound on the number of zeros of simple continuous non-polynomials

Let $[vf(p)-g(p)]$ be a degree $n$ polynomial, one of whose roots are $p = F(v)$.

Note: We said $F(v)$ is one of the $n$ roots. Hence, it is a single valued and continuous function of $v$.

Let us now declare another polynomial: $L(p,v) = \sum_{i=0}^{n}(vA_{i} + B_{i}) p^{i}$. When expressed in terms of $v$ alone, we have:

$L(v) = \sum_{i=0}^{i}(vA_{i} + B_{i})[F(v)]^{i}$

which is no more a polynomial in $v$. However, it is certainly a continuous function of $v$. Moreover, if we assume the function to be non-constant, can we bound the number of zeros of this function ?

I would be equally happy to know how small of an interval does one need to choose about any of its zeros, to find a point at which $L(v)$ is non-zero ?

(All the constants here are real numbers.)

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