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Journal Articles Discrete and Computational Geometry Year : 2015

On the intersection of a sparse curve and a low-degree curve: A polynomial version of the lost theorem

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Abstract

Consider a system of two polynomial equations in two variables: $$F(X,Y)=G(X,Y)=0$$ where $F \in \rr[X,Y]$ has degree $d \geq 1$ and $G \in \rr[X,Y]$ has $t$ monomials. We show that the system has only $O(d^3t+d^2t^3)$ real solutions when it has a finite number of real solutions. This is the first polynomial bound for this problem. In particular, the bounds coming from the theory of fewnomials are exponential in $t$, and count only nondegenerate solutions. More generally, we show that if the set of solutions is infinite, it still has at most $O(d^3t+d^2t^3)$ connected components. By contrast, the following question seems to be open: if $F$ and $G$ have at most $t$ monomials, is the number of (nondegenerate) solutions polynomial in $t$? The authors' interest for these problems was sparked by connections between lower bounds in algebraic complexity theory and upper bounds on the number of real roots of ''sparse like'' polynomials.
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Dates and versions

ensl-00871315 , version 1 (09-10-2013)
ensl-00871315 , version 2 (23-07-2014)

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Pascal Koiran, Natacha Portier, Sébastien Tavenas. On the intersection of a sparse curve and a low-degree curve: A polynomial version of the lost theorem. Discrete and Computational Geometry, 2015, pp.16. ⟨ensl-00871315v2⟩
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