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Let P^n denote the n-dimensional projective space defined over the algebraicclosure of a finite field F_q, let V contained P^n be a complete intersectiondefined over F_q of dimension r and singular locus of dimension at most s, andlet \pi:V-->P^{s+1} be a "generic" linear mapping. We obtain an effectiveversion of the Bertini smoothness theorem concerning \pi, namely an explicitupper bound of the degree of a proper Zariski closed subset of P^{s+1} whichcontains all the points defining singular fibers of \pi. For this purpose wemake essential use of the concept of polar variety associated to the set ofexceptional points of \pi. As a consequence of our effective Bertini theorem weobtain results of existence of smooth rational points of V, namely conditionson q which imply that V has a smooth q-rational point. Finally, for s=r-2 ands=r-3 we obtain estimates on the number of q-rational points of V, and wediscuss how these estimates can be used in order to determine the average valueset of "small" families of univariate polynomials with coefficients in F_q.

Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field