Counterexamples to the Connectivity Conjecture of the Mixed Cells

Transcription

1 Dscrete Comput Geom 2: Dscrete & Computatonal Geometry 998 Sprnger-Verlag New York Inc. Counterexamples to the Connectvty Conjecture of the Mxed Cells T. Y. L and X. Wang 2 Department of Mathematcs Mchgan State Unversty East Lansng MI USA 2 Department of Mathematcs Unversty of Central Arkansas Conway AR 7235 USA Abstract. In [4] a conjecture concernng the connectvty of mxed cells of subdvsons for Mnkowsk sums of polytopes was formulated. Ths conjecture was n fact orgnally proposed by Pedersen [3]. It turns out that a postve confrmaton of ths conjecture can substantally speed up the algorthm for the dynamcal lftng developed n [4]. In the mean tme when the polyhedral method s used for solvng polynomal systems by homotopy contnuaton methods [2] the postveness of ths conjecture also plays an mportant role n the effcency of the algorthm. Very unfortunately we found that ths conjecture s naccurate n general. In Secton a counterexample s presented for a general subdvson. In Secton 2 another counterexample shows that even restrcted to regular subdvsons nduced by lftngs ths conjecture stll fals to be true.. Counterexample for General Subdvsons For m n let A = A...A m be a sequence of fnte subsets of R n whose unon affnely spans R n.forq = conva the convex hull of A =...m ther Mnkowsk sum s defned by Q + +Q m ={x + +x m x Q for =...m}. By a cell of A we mean a tuple C = C...C m of subsets C A for =...m. Defne typec := dmconvc...dmconvc m N m convc := convc + +C m R n. The research of the frst author was supported n part under NSF Grant DMS and a Guggenhem fellowshp. The second author s research was supported n part by a UCA summer stpend.

2 56 T. Y. L and X. Wang Here N s the set of natural numbers. Cells of the same type are called the mxed cells of that type. A face of a cell C s a subcell F = F...F m where F C and some lnear functonal α R n v attans ts mnmum over C at F for =...m. Such an α s called an nner normal of F. IfF s a face of C then convf s a face of the polytope convc for =...m. Defnton. such that A fne mxed subdvson of A s a collecton S ={C...C k } of cells a dmconvc j = n for all j =...k b convc j convc l s a proper common face of convc j and convc l when t s nonempty for j l c k j= convc j = conva d for j =...k wrte C j = C j j...c m ; then each convc j s a smplex of dmenson #C j and for each j dmconvc j + + dmconvc m j = n. Remark. In fact the notaton A = A...A m s just a shorthand abstracton of conva = conva + +A m. So s a cell C = C...C m n the subdvson of A where C A for =...m; namely t means convc = convc + +C m. 2 Later when we draw fgures for A = A...A m or subdvsons of A they actually represent and 2. For a gven fne mxed subdvson S of A = A...A m let S ={B...B l } S be the set of all mxed cells n S havng the same type k...k m wth k > for all =...m. Wrte B j = B j j...b m for j =...l. Two cells D and E n S are sad to be connected f there exsts a sequence of cells {B j...b jd } n S such that a B j = D and B jd = E and b dmconvb j p convb j p+ k for all =...m and p =...d. It s conjectured n [3] and [4] that all mxed cells of the same type n S are connected. The followng example shows that ths conjecture may not hold n general. Example. Let A = { a = b = 2 c = d = 2

3 Counterexamples to the Connectvty Conjecture of the Mxed Cells 57 Fg. and A 2 = { e = f = g =. Q = conva Q 2 = conva 2 and Q + Q 2 are shown n Fg.. Consder the collecton of cells C = C C 2 C 3 C 4 C 5 where C ={a d e f C 2 ={b d c e C 3 ={b c e g C 4 ={b e f g C 5 ={a b d f. Clearly C s a fne mxed subdvson of A = A A 2 and cells C and C 3 are the only cells of type. They are not connected because a d b c =. 2. Counterexample for Regular Subdvsons For A = A...A m as n Secton each A s a fnte subset of R n choose realvalued functons ω : A R for =...m. The m-tuple ω = ω...ω m s called a lftng functon on A. We say that ω lfts A to ts graph  ={qω q : q A } R n+. Ths noton s extended n the obvous way:  = Â... m Q = convâ Q = Q + + Q m etc.

4 58 T. Y. L and X. Wang Let S ω be the set of cells C of A whch satsfy a dmconvĉ = n and b Ĉ s a facet face of dmenson n ofaˆ whose nner normals α R n+ v have postve last coordnate. In other words convĉ s a facet of the lower hull of ˆQ. It s known [] that S ω s a subdvson of A and for a generc lftng functon ω S ω s always a fne mxed subdvson. A subdvson nduced by a lftng functon s called a regular subdvson. When the polyhedral method s used for solvng polynomal systems by homotopy contnuaton methods [2] the lftng functon ω provdes a nonlnear homotopy for the algorthm and ts nduced subdvson S ω for the supports of the polynomals plays an essental role for the whole process of the algorthm. For the effcency of the algorthm t s very desrable that the mxed cells of a fxed type of ths subdvson are connected. In fact Pedersen showed [3] that when n = 2 mxed cells of a fxed type are ndeed connected for regular subdvsons. The dea of the proof was sketched n [4]. Unfortunately the followng example shows ths conjecture s ncorrect when n = 3. Example 2. Let A = a = A 2 = A 3 = b = c = a 2 = b 2 = c 2 = a 3 = b 3 = c 3 = c 4 = and ts lftng  =  2 =  3 = { a { b { c a2 b2 c2 a3 b3 c3 4 c4. The regular subdvson of A = A A 2 A 3 nduced by the above lftng s shown n Fg. 2. In ths fne mxed subdvson there are eght cells they are: C ={a b b 2 b 3 c 2 c 4 C 2 ={a b 2 b 3 c 2 c 3 c 4 C 3 ={a b b 3 c c 2 c 4 C 4 ={a a 3 b 2 b 3 c 3 c 4 C 5 ={a a 2 b b 3 c c 2 C 6 ={a a 3 b 3 c 2 c 3 c 4 C 7 ={a a 2 b 3 c c 2 c 4 C 8 ={a a 2 a 3 b 3 c 2 c 4.

5 Counterexamples to the Connectvty Conjecture of the Mxed Cells 59 Fg. 2 see Fg. 3. Clearly there are two cells of type they are C 4 and C 5.Onthe other hand α = 2 and α 2 = are the nner normals of Ĉ 4 and Ĉ 5 respectvely. Obvously C 4 and C 5 are not connected snce c c 2 c 3 c 4 =. Remark. The dynamc lftng algorthm n [4] constructs placeable subdvsons whch only requre mxed cells of a certan type to be fne. We trust that counterexamples to the connectvty of mxed cells n placeable regular subdvsons can also be constructed. 2

6 52 T. Y. L and X. Wang Fg. 3

7 Counterexamples to the Connectvty Conjecture of the Mxed Cells 52 References. I. M. Gel fand M. Kapranov and A. Zelevnsky Multdmensonal Determnants Dscrmnants and Resultants Brkhäuser Boston B. Huber and B. Sturmfels A polyhedral method for solvng sparse polynomal systems. Math. Comp. 6422: P. Pedersen Personal communcaton at the occason of the AMS IMS SIAM Summer Research Conference on Contnuous Algorthm and Complexty Mount Holyoke College South Hadley MA June J. Verschelds K. Gatemann and R. Cools Mxed volume computaton by dynamcal lftng appled to polynomal system solvng Dscrete Comput. Geom. 6: Receved September and n revsed form February