We obtain new necessary conditions for an n-dimensional semialgebraic subset of R '~ to be a polynomial image of I~ n. Moreover, we prove that a large family of planar bidimensional semialgebraic sets with piecewise linear boundary are images of polynomial or regular maps, and we estimate in both cases the dimension of their generic fibers. Isr. J. Math. that several not necessarily convex sets with piecewise linear boundary are polynomial images, and we estimate their invariants p and r. The paper ends with the formulation of some selected open questions.Most of the results of this work are also true if we change the field it( of the real numbers by any other real closed field. This generalization is quite straightforward, and we will not enter here into its details. We refer the reader to [BCR].
ACKNOWLEDGEMENTS:The authors thank Prof. J. M. Ruiz for many fruitful discussions during the preparation of this work. We also thank Profs. J. Guti~rrez and D. Sevilla for pointing out several classical results used in Section 2.
One-dimensional polynomial imagesIt is a difficult question to decide under what conditions a polynomial map f: ]~n __+ ]~m factors through R d for d = dim(f(~n)). We prove that this always happens for d = 1, and that fact helps us to get a better understanding of onedimensional polynomial images. We also see that for d > 2 there exist examples of maps that do not factorize. PROPOSITION 2.1: Let f = (fl,..., fro): ~n _.4 ~m be a polynomial map whose image has dimension 1. Then f factors polynomially through ~, that is, there exist polynomial maps g: ~n _~ I~ and h: ~ --+ ~m such that f = h o g. Proof: Let ]~ ~---]~(fl,''', fra) be the smallest subfield of the field of rational functions ]R(Xl,.. •, xn) in n variables that contains IR and fl,. ,., ,fro. Note that tr. deg(FIR ) = dim im f = 1, and that some of the fi's are not constant. Then, by [N, §13], [Sch, §3. Thm. 4], there exists a polynomial g E R[Xl,..., xn] such that F = ]l((g).Next, we are seeking polynomials hl,...,hm E ~t] such that hi(g) = fi. For that, since fi E F = ~(g), we have fi --Pi(g)/Qi(g) for some coprime polynomials Pi, Qi E ~[t]. By Bezout's lemma, we can write 1 = PiAi + QiBi for some A~, B~ E If(It]. Substituting the variable t by g we get the polynomial identity i = Pi(g)Ai(g) + Qi(g)Bi(g) = Qi(g)fiAi(g) + Qi(g)Bi(g) = Qi(g)(fiA~(g) + B~(g));Vol. 153, 2006 POLYNOMIAL AND REGULAR IMAGES OF •n 65