We investigate properties of finite relational structures over the rea
ls expressed by first-order sentences whose predicates are the relatio
ns of the structure plus arbitrary polynomial inequalities, and whose
quantifiers can range over the whole set of reals. In constraint progr
amming terminology, this corresponds to Boolean real polynomial constr
aint queries on finite structures. The fact that quantifiers range ove
r all reals seems crucial; however, we observe that each sentence in t
he first-order theory of the reals can be evaluated by letting each qu
antifier range over only a finite set of real numbers without changing
its truth value. Inspired by this observation, we then show that when
all polynomials used are linear, each query can be expressed uniforml
y on all finite structures by a sentence of which the quantifiers rang
e only over the finite domain of the structure. In other words, linear
constraint programming on finite structures can be reduced to ordinar
y query evaluation as usual in finite model theory and databases. More
over, if only ''generic'' queries are taken into consideration, we sho
w that this can be reduced even further by proving that such queries c
an be expressed by sentences using as polynomial inequalities only tho
se of the simple form chi < y.