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In this paper, we consider a problem of counting multiplicities. We fix acounting function of multiplicity of rational points in a hypersurface of aprojective space over a finite field, and we give an upper bound for the sumwith respect to this counting function in terms of the degree of thehypersurface, the dimension and the cardinality of the finite field. This upperbound gives a description of the complexity of the singular locus of thishypersurface. In order to obtain this upper bound, we introduce a notion calledintersection tree by intersection theory. We construct a sequence ofintersections, such that the multiplicity of a singular rational point is equalto that of one of the irreducible components in these intersections. Themultiplicities of these irreducible components constructed above are bounded bytheir multiplicities in the intersection tree.

Comptage des multiplicit\'es dans une hypersurface sur un corps fini