The measure for a family of hypersurfaces with multiple components

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1 Int. J. Contemp. Math. Sci., Vol., 2006, no. 0, The measure for a family of hypersurfaces with multiple components Giuseppe Caristi University of Messina Faculty of Economics, 9822 Me) Italy. gcaristi@dipmat.unime.it Giovanni Molica Bisci University of Reggio Calabria Faculty of Engineering, DIMET. Via Graziella Feo di Vito) I-8900 Reggio Calabria Italy. giovanni.molica@ing.unirc.it Abstract In the real affine 5-dimensional space A 5, we show the measurability of the family of reducible hypersurfaces of type S = p n... pn 5 5, where the components p i are hyperplanes passing trough a fixed point and multiplicity n i N. Mathematics Subect Classification: 60D05, 52A22. Keywords: Geometric probability; stochastic geometry; random sets; random convex sets and integral geometry. Introduction Problems of measurability in the affine spaces have been studied by many authors, for various families of varieties and different kinds of geometric configurations see [],[2] and [5],[6]). Let X n be a space of dimension n, for example the affine or the proective space over the real field and G r : x i = f ix,..., x n ; a,..., a r ), i =,..., n), ) a Lie group of transformations on X n with a,..., a r independent set of essential parameters for G r.

2 464 G. Caristi and G. Molica Bisci We assume that the identity of the group G r is determined by a =... = a r = 0. A function φ, solution of the system of partial differential equations n i= x i ξ i x,..., x n )φx,..., x n )) = 0, 2) where ξ i x,..., x n )= x i, a a=0 =,..., r), is called invariant integral function for the group G r. M. Stoka [8] defines measurable a group G r with a unique invariant integral function φ, uptoa constant factor. Let V q be a family of p-dimensional varieties defined by V q : F x,..., x n ; α,..., α q )=0,..., F n q x,..., x n ; α,..., α q ) = 0 where α s R, s =,..., q are essential independent parameters and F k, k =,..., n p) are analytic functions on its domains. Following the definitions and the proof in [8], if G r is the maximal invariant group of V q one has an associated group of transformations H r : α h = ϕ hα,..., α q ; a,..., a r ), h =,..., q), and G r and H r are isomorphic. If we suppose that H r is measurable of invariant integral function φα,..., α q ), the expressions μ Gr V q )= φα,..., α q )dα... dα q, Fα) where Fα) ={α,..., α q ) R q F k x; α,..., α q )=0,k=,..., n p} and φα,..., α q ) dα... dα q, are called respectively the measure of the family V q and the invariant density respect to the group G r of the family V q. Hence, by definition, V q is measurable if there exists a unique non-zero function φ. Now let A 5 be the five dimensional affine space over the real field) of coordinates x,..., x 5. The goal of this paper is to show the measurability of the family of reducible hypersurfaces in A 5 of type S = p n... p n 5 5, where the components p i are hyperplanes passing trough a fixed point and multiplicity n i N.

3 The measure for a family of hypersurfaces with Main Results Without loss of generality, we can suppose that the fixed point for the components of the surface S is the origin of coordinates. Hence the generic element of the family is the surface S of equation S : = x + A ) x A 4) x 5 ) n =0, n N), where deta l) ) 0, with l =0,..., 4; =,..., 5 and A 0) :=, for 5. The parameters space of this family is of dimension 20 and coordinates R with i 4 and =,..., 5. The maximal group of invariance of the family is G 5 2 : x r = with α s r R and detα s r) 0. s= αr s x s, r =, 2,..., 5 3) Acting by G 5 2 on S we obtain S : = [ ] R, with x +[A ) ] x [A 4) ] x 5 ) n =0, n N), H 5 2 : [ ] = α + 4 l= A l) α + 4 l= A l) α l αl, 4) where i 4; =,..., 5 and α + 4 l= A l) α l 0, =,..., 5. Theorem 2. The family of reducible hypersurfaces in A 5 element of equation of generic S : = x + A ) x A 4) x 5 ) n =0, n N), where deta l) ) 0l =0,..., 4; =,..., 5, A 0) :=, for 5) is measurable and its unique invariant integral function is given by φ = l) 5. det A i

4 466 G. Caristi and G. Molica Bisci Proof : By direct calculations we give the following Deltheil s system: 4 = i= = 20φ, = =0, i =,..., 4 = = 5φ, And finally = =0, i,k =,..., 4; k i. = = h) 2 A + k) 2 A + = = for every pair h, k) with h k =,..., 4. = 5 = 5 = = φ, φ, The unique non-zero solution up to a constant factor) is φ = l) 5. det A i Remark The three dimensional and reduced case is investigated by Santoro in [5]. References [] P. Dulio, M. Petriccione, Sulla misurabilità delle famiglie dei sistemi di k iperpiani passanti per un punto fisso dello spazio affine A 7, Atti Acc. Peloritana dei Pericolanti Classe S.M.F.N. Vol.LXXI 993). [2] P. Dulio, M. Petriccione, Sulla misurabilità delle famiglie dei sistemi di due e tre iperpiani passanti per un punto fisso di A 4, Atti Acc. Peloritana dei Pericolanti Classe S.M.F.N. Vol.LXXI 993).

5 The measure for a family of hypersurfaces with [3] A. Duma, M. Stoka, On the measurability of the conic sections family in the proective space P 3, Rend. Circ. Mat. Palermo, Serie II, Tomo LI 2002), [4] G. Failla, G. Molica Bisci, On the measurability of the conic sections family in the proective space P 4, to appear Boll. U.M.I. [5] G. Santoro, Sulla misurabilità della famiglia di superfici cubiche spezzate in tre piani passanti per un punto fisso, Atti Acc. Peloritana dei Pericolanti Classe S.M.F.N., Vol.LXVIII, 990). [6] G. Santoro, Sulla misurabilità di una famiglia di superfici cubiche dello spazio affine A 3, Atti Acc. Peloritana dei Pericolanti Classe S.M.F.N., Vol.LXVIII, 990). [7] M. Stoka, Masura familiior de varietati din spatiul E 3, St. Cerc. Mat., IX, 958), [8] M. Stoka, Géométrie Intégrale, Mém. Sci. Math., fasc. 65, Gauthier- Villars, Paris, 968. [9] M. Stoka, On the measurability of the conic sections family in the proective space P n, Boll. U.M.I. 8) 7-B 2004), Received: June 6, 2006