Elimination of parasitic solutions in theory of flexible polyhedra
aa r X i v : . [ m a t h . M G ] F e b ELIMINATION OF PARASITIC SOLUTIONS IN THEORYOF FLEXIBLE POLYHEDRA
I. KH. SABITOV AND D. A. STEPANOV
Abstract.
The action of the rotation group SO (3) on systems of n points in the 3-dimensional Euclidean space R induces naturally anaction of SO (3) on R n . In the present paper we consider the followingquestion: do there exist 3 polynomial functions f , f , f on R n suchthat the intersection of the set of common zeros of f , f , and f with each orbit of SO (3) in R n is nonempty and finite? Questions of thiskind arise when one is interested in relative motions of a given set of n points, i. e., when one wants to exclude the local motions of the system ofpoints as a rigid body. An example is the problem of deciding whether agiven polyhedron is non-trivially flexible. We prove that such functionsdo exist. To get a necessary system of equations f = 0, f = 0, f = 0,we show how starting by choice of a hypersurface in CP n − containingno conics, no lines, and no real points one can find such a system. Introduction
For many problems in mathematics and physics one needs to study someproperties of a system of points which depend only on the distances betweensome or all the pairs of points. In such a case the corresponding propertiesshould be invariant under the motion of the points as particles of a rigidbody or under an orthogonal change of coordinates. At the same time sucha motion leads to different positions of the system in space, thus posinga question on geometric or physical identity of the new system with theinitial one. For example, if we look for a simplicial polyhedron with givencombinatorial structure and known edge lengths, then the coordinates of itsvertices can be obtained as solutions of the system of equations(1) ( x i − x j ) + ( y i − y j ) + ( z i − z j ) = l ij , where the pair ( i, j ) varies over all edges of the polyhedron with knownlength l ij . However, having found two different solutions of these equations,we do not know a priori whether they give polyhedra that differ only bya continuous motion as rigid bodies or these polyhedra are isometric andnoncongruent ones. Note that we call two polyhedra isometric if they havethe same combinatorial structure and the corresponding edges are equal,but the polyhedra are not necessarily related by an isometry of the ambientEuclidean space. This difficulty is usually formally avoided by saying that“solutions are considered up to a motion as a rigid body” or using the formaloperation of factorization of the set of polyhedra by the isometry group ofthe affine Euclidean space R . But this trick does not give a direct answerto the question whether two close solutions are related by an isometry of R or by a non-trivial flexion with rigid faces (that is by a continuous family of solutions of (1); one uses the term bending too) and in practice one has tosomehow check this. In his papers (see, e.g., [9], [10], [11]) the first authorsuggested to replace the factorization operation by adding to system (1)several new equations which should automatically exclude the “parasitic”solutions. For instance, it is always possible to suppose that the origin ofcoordinates is placed in the center of mass of the given system of materialpoints and then to adjoin to (1) three new equations(2) X j x j = 0 , X j y j = 0 , X j z j = 0(all the masses are supposed to be equal), thus eliminating the parasiticsolutions related to a parallel translation. Further, to exclude solutions thatare obtained from a given one by a rotation around the origin (that is, bya transformation from the group SO (3)), we could impose an additionalcondition that the system of coordinates is chosen in accordance with theprincipal axes of inertia of given points. This choice of coordinates is equiv-alent to adjoining three more equations X j x j y j = 0 , X j y j z j = 0 , X j x j z j = 0to system (1). But these new equations work well only under the assumptionthat X j x j = X j y j , X j y j = X j z j , X j x j = X j z j , otherwise the system of points admits rotations around one of the coordinateaxes. At the same time we look for equations that would exclude rotationsfor any initial position of the points. Another variant of screening for ”true”flexion consists in fixation of one of the non-degenerate triangular faces. Inturn, this method is not universal also since for some collections of points thegiven face can be degenerated so we should know in advance the existenceof a non-degenerated face.One more example where one needs to somehow fix a system of points isthe problem of recovering of positions of points from the given distances be-tween each pair of them, as in the well known Lennard-Jones Problem fromphysics. In Lennard-Jones Problem one seeks a minimum of a functionalwhich is a function of the distances between points. If, say, the necessarydistances have been determined, then to recover the positions of points ofthe initial system one has to choose a unique configuration which suits inthe best way some additional requirements. Thus again one has to lookfor solutions of a system analogous to (1) and eliminate the repetition ofcopies. Another important example is the classical n -body problem, wherethe quotient R n /SO (3) (referred to as the shape space ) gives some inter-esting insights, see [8].Now let us formulate our problem and results more precisely. Let P , . . . , P n be a system of points in the affine space R m with the standard Euclideanstructure. Some or even all of the points of the system can coincide, that is,possess equal coordinates. Let G be a group of isometries of R m . If ρ is anelement of G , then by ρP j we denote the image of the point P j under theaction of ρ . Note that the system { P j } nj =1 can be represented by a single LIMINATION OF PARASITIC SOLUTIONS 3 point M in the Euclidean space R mn = R m × · · · × R m , thus we have anatural diagonal action of the group G on R mn . The orbit of the point M in R mn under this action will sometimes be referred to also as the orbit ofthe system { P j } nj =1 .Let f = ( f , . . . , f k ) be a collection of real valued functions on R mn .Keeping in mind the splitting of R mn as a product of n copies of R m , weshall consider each of the functions f , . . . , f k also as a function of n pointsin R m . We shall say that the system of points { P j } nj =1 ⊂ R m admits areduction by G with respect to f , if there exists an element ρ ∈ G such that f l ( ρP , . . . , ρP n ) = 0 for each l = 1 , . . . , k . In other words, the variety X f = { M ∈ R mn | f ( M ) = · · · = f k ( M ) = 0 } has a nonempty intersection with the orbit of the system { P j } nj =1 . We shallsay that the system { P j } nj =1 is fixed by f with respect to G , if the orbitof { P j } nj =1 has no more than finite number of intersection points with thevariety X f . For example, for m = 3 and G the group of parallel translationsof R , each system of points { P j } nj =1 ⊂ R admits a reduction with respectto and is fixed by the functions on the left hand side of (2). Instead ofthe collection of functions f , we sometimes say that a system of pointsadmits a reduction with respect to (or is fixed by) the system of equations f = · · · = f k = 0.In Section 2 we study the case m = 2, that is, the case of the Euclideanplane R , and G = SO (2) the group of plane rotations around the origin. Weconsider the canonical representation of SO (2) as the group of orthogonal 2by 2 matrices with determinant one which act on R by left multiplication.We show that each system of points { P j } nj =1 = { ( x ◦ j , y ◦ j ) } nj =1 on the planeadmits a reduction with respect to and in the same time is fixed by thefunction A ( x , y , . . . , x n , y n ) = ( x n y n ) n − + · · · + ( x y ) + x y proposed by our untimely deceased colleague A. V. Astrelin. Note that if atleast one of the points P j of a system { P j } nj =1 is different from the origin,then the orbit of such a system in R n is homeomorphic to the circle S ≃ SO (2).Now let { P j } nj =1 = { ( x ◦ j , y ◦ j , z ◦ j ) } nj =1 be a system of points in the 3-dimensional Euclidean space R , and G = SO (3) be the group of spacerotations around the origin in its canonical representation. In Section 3, weprove our main result. Theorem 1.1.
Let F ( w , . . . , w n ) be a homogeneous polynomial of somedegree d such that the hypersurface X F defined by F = 0 in the complex projective space CP n − contains no conics, no lines, and no real points. Let H ( w , . . . , w n ) = w p + w p + · · · + w p n n , p > p > · · · > p n , where w j = x j + iy j are complex variables. Then each system of n points in R admits a reduction by the group SO (3) with respect to the functions f = Re F, f = Im F, f = Im H, considered as functions on R n , and at the same time is fixed by f , f , f with respect to SO (3) . I. KH. SABITOV AND D. A. STEPANOV
For the diagonal action of SO (3) on R n , the orbit of the system { P j } nj =1 (that is, the orbit of M in R n ) is homeomorphic to the real 3-dimensionalprojective space RP ≃ SO (3) whenever the points O , P , . . . , P n are notcollinear, where O is the origin (see Lemma 3.1). If the points O, P , . . . , P n are collinear but at least one of the points P , . . . , P n is different from theorigin, then the orbit is homeomorphic to the 2-dimensional sphere S , fi-nally, the orbit is reduced to a single point if all the points coincide with theorigin.The full group of affine isometries of R is generated by the subgroup ofparallel translations and by the full orthogonal group O (3). But the orbitof a system of points under the action of O (3) is a union of no more than 2orbits of SO (3), thus we get the following corollary. Corollary 1.2.
Let G be the full group of affine isometries of R . Then eachsystem of points in R admits a reduction by G with respect to functions of which are defined in (2) and the other in Theorem 1.1, and at the sametime is fixed by these functions. Corollary 1.2 can be applied in the theory of flexible polyhedra. Assumenow that { P j } nj =1 are the vertices of a polyhedron P in R and we areinterested whether polyhedron P is flexible. Consider the system of algebraicequations (1) fixing the lengths of edges of P and adjoin to this system 6 moreequations, namely, 3 equations (2) and 3 more equations f = f = f = 0for f , f , f from Theorem 1.1. Let us call so obtained system the extendedsystem of the polyhedron P . Corollary 1.3. (a) A polyhedron P with coordinates of vertices representedby a point M = ( x ◦ , y ◦ , z ◦ , . . . , x ◦ n , y ◦ n , z ◦ n ) in R n satisfying the extendedsystem is not flexible if and only if there exists a neighborhood U of M in R n such that M is the only solution to the extended system of equationsin U . (b) All polyhedra isometric to the given polyhedron P are not flexibleif and only if the extended system of equations of P has only finite numberof solutions. To facilitate the correct understanding of Corollary 1.3 recall that, forexample, together with the flexible Bricard octahedron of the first typethere exists an isometric to it continuously rigid octahedron. Thus, in theneighborhoods of the corresponding points in R the solution set to theextended system has different structure. Remark . As the reader has perhaps noticed, the functions f , f , f in Theorem 1.1 depend only on the projections of the points P j to thecoordinate plane Oxy . Evidently an analogue of Theorem 1.1 is valid forthe cases when the planes
Oxz and
Oyz are selected.
Remark . Existence of hypersurfaces X F with the properties stated inTheorem 1.1 follows from general results of algebraic geometry. For example,we can quote the main result of [1] which says that a generic hypersurfaceof degree r in CP n has no rational curves as soon as r ≥ n −
1. Thisgives the estimate 2 d ≥ n − r > (3 n + 2) ([3, Theorem 1.1]). We insist on even degree 2 d to be able LIMINATION OF PARASITIC SOLUTIONS 5 to get a hypersurface with no real points. Indeed, for this we can take ageneric hypersurface in a small neighborhood (in the space parameterizingdegree 2 d hypersurfaces in CP n ) of the Fermat hypersurface w d + w d + · · · + w dn = 0 . From computational perspective, it is desirable to choose polynomial F with,let us say, “simple” coefficients, i.e., the hypersurface X F to be defined over Q or some of its finite algebraic extensions. Definitely, having chosen aparticular hypersurface that seems likely to fulfill our conditions, it shouldbe possible to check by a computer whether it really does. But we are notaware of any general method that would give such a hypersurface for all n .We know two published works where a problem similar to ours was con-sidered. The first is paper [2] of R. Connelly, where at the end the authorpresents an equation restricting a motion of a system of points in R insuch a way that if their motion is a part of a rigid motion of all of R ,then the system in reality is fixed. This result is, however, very differentfrom ours because Connelly examines not the question of flexibility of agiven polyhedron, but only the question of triviality of a given flexion ofa polyhedron (or a system of points). In [4, Section 2.3.4, Exercise (d’)],M. Gromov proposes to the reader to prove that if an algebraic foliationof R n into codimension q leaves is given, then there exists a q -dimensionalalgebraic subset of R n intersecting all the leaves of the foliation. The mainidea, as can be guessed from the preceding discussion in [4], is to choose asufficiently generic polynomial f on R n and to consider the locus of criticalpoints of all the restrictions of f to each leaf of the foliation. Our resultsdo not, however, follow from this exercise since, in Gromov’s approach, itis not clear why that q -dimensional algebraic subset must be given exactlyby q equations (i.e., is a complete intersection in the language of algebraicgeometry); also, the question whether the number of intersection points ofthe algebraic subset with each leaf is finite (the question of fixation in ourterminology) is not discussed in [4].2. Reduction and fixation of a system of points in the plane
A function A : R n → R , where A ( x , y , . . . , x n , y n ) = ( x n y n ) n − + · · · + ( x y ) + x y , will be called the plane Astrelin function . Theorem 2.1.
Each system { P j } nj =1 of points in the plane admits a reduc-tion by the group SO (2) with respect to the plane Astrelin function A andat the same time is fixed by A with respect to SO (2) .Proof. If all the points of the given system are concentrated in the origin,then the orbit of the system under the action of SO (2) is reduced to a singlepoint and the theorem is obviously true in this case. Thus in the rest of theproof we assume that at least one of the points P j , j = 1 , . . . , n , is differentfrom the origin. I. KH. SABITOV AND D. A. STEPANOV
The reduction with respect to the Astrelin function is easy to prove. If σ is a rotation by 90 ◦ and P j = ( x ◦ j , y ◦ j ), then σP j = ( − y ◦ j , x ◦ j ) and A ( σP , . . . , σP n ) = − A ( P , . . . , P n ) . Since the group SO (2) is pathwise connected, there exists an angle ϕ , 0 ≤ ϕ ≤ ◦ , such that the rotation ρ ϕ by this angle reduces the system: A ( ρ ϕ P , . . . , ρ ϕ P n ) = 0 . The proof of fixation starts with the following parametrization of SO (2): SO (2) = (cid:26)(cid:18) cos θ − sin θ sin θ cos θ (cid:19) (cid:12)(cid:12)(cid:12) θ ∈ R (cid:27) . The restriction of the plane Astrelin function to an orbit is given by thefunction A ( θ ) = [ 12 (( x ◦ n ) − ( y ◦ n ) ) sin 2 θ + x ◦ n y ◦ n cos 2 θ ] n − + · · · ++ [ 12 (( x ◦ ) − ( y ◦ ) ) sin 2 θ + x ◦ y ◦ cos 2 θ ] ++ [ 12 (( x ◦ ) − ( y ◦ ) ) sin 2 θ + x ◦ y ◦ cos 2 θ ] , which is analytic as a function of θ . If the equation A ( θ ) = 0 had infiniteset of solutions on the interval [0 , π ], then the function A ( θ ) would vanishidentically, and thus all of its Fourier coefficients would vanish too. TheFourier coefficient with the biggest number can be found from the expansioninto Fourier series of the first summand A n − ( θ ) = [ 12 (( x ◦ n ) − ( y ◦ n ) ) sin 2 θ + x ◦ n y ◦ n cos 2 θ ] n − . We can assume that P n = O , thus this first summand is nonzero. Considera representation A n − ( θ ) = (cid:16) α n e iθ + e − iθ β n e iθ − e − iθ i (cid:17) n − = ( γ n e iθ + ¯ γ n e − iθ ) n − , where α n = x ◦ n y ◦ n , β n = 12 (( x ◦ n ) − ( y ◦ n ) ) , γ n = α n − iβ n . Note that γ n = 0. Further, by Newton binomial formula( γ n e iθ + ¯ γ n e − iθ ) n − = n − X k =0 C k n − γ kn ¯ γ n − − kn e i (4 k − n +2) θ . The leading Fourier coefficient a n − + ib n − is calculated by the formula a n − + ib n − = 1 π Z π n − X k =0 C k n − γ kn ¯ γ n − − kn e i (4 k ) θ dθ, and from this entire sum only the summand with k = 0 survives. Therefore, a n − + ib n − = 2¯ γ n − n = 0 , that is A ( θ )
0. We get a contradiction that proves the theorem. (cid:3)
LIMINATION OF PARASITIC SOLUTIONS 7
Example . Consider the square with initial position of vertices P = (1 , P = (0 , P = ( − , P = (0 , − x j = cos (cid:16) ϕ + ( j − π (cid:17) , y j = sin (cid:16) ϕ + ( j − π (cid:17) , j = 1 , , , , we reduce Astrelin’s equation x y + ( x y ) + ( x y ) + ( x y ) = 0to the equation12 sin 2 ϕ −
18 sin ϕ + 132 sin ϕ − ϕ = 0 , or 12 sin 2 ϕ − sin ϕ sin ϕ = 0 . It follows that ϕ = πk/
2, i.e., the square is fixed by Astrelin function withrespect to rotation in its initial position and in the positions which differfrom the initial one by an angle multiple to π/
2. In the same time thisquadrangle admits a deformation(3) x ( t ) = 1 − t, y ( t ) = 0; x ( t ) = 0 , y ( t ) = √ t − t ; x ( t ) = − t, y ( t ) = 0; x ( t ) = 0 , y ( t ) = −√ t − t keeping the lengths of edges, the center of mass, and satisfying Astrelin’scondition. So then we can affirm that the deformation (3) is a nontrivialflexion of the square (compare with (a) of Corollary 1.3). At the sametime, the fact that the same deformation satisfies the classical condition P j x j y j = 0 does not ensure its non-triviality. Remark . The hypersurface X A = { ( x , y , . . . , x n , y n ) ∈ R n | A ( x , y , . . . , x n , y n ) = 0 } ⊂ R n is singular at points where x = y = 0 and for each j ≥ x j = 0 or y j = 0.But even at its smooth points X A can have non-transversal intersectionswith orbits of SO (2). Indeed, the tangent vector to the orbit of SO (2) at apoint ( x , y , . . . , x n , y n ) ∈ R n is ( − y , x , . . . , − y n , x n ). Thus, this point isa non-transversal intersection point of X A with an orbit if and only if ( ( x n y n ) n − + · · · + ( x y ) + x y = 0 , (2 n − x n y n ) n − ( x n − y n ) + · · · + 3( x y ) ( x − y ) + ( x − y ) = 0 . An example of a non-zero solution to this system of equations can be ob-tained by letting x j = y j for all j = 2 , . . . , n and x = − y , so that the secondequation is satisfied, then choosing arbitrary non-zero values for x j = y j for j = 2 , . . . , n , and, finally, choosing x = − y so that the first equation isalso satisfied. I. KH. SABITOV AND D. A. STEPANOV Reduction and fixation of a system of points in space
The main purpose of this section is to prove Theorem 1.1. But we startwith a lemma describing the non-degenerate orbits of the natural action of SO (3) on R n , n ≥
2. This lemma is not needed for the proof of Theo-rem 1.1, however, we believe that it is useful for better understanding of thesituation under study.
Lemma 3.1.
Assume that not all of the points of a system { O } ∪ { P j } nj =1 ⊂ R , n ≥ , are collinear. Then the orbit in R n of the system { P j } nj =1 underthe natural action of the group SO (3) is homeomorphic (in fact, diffeomor-phic) to the Lie group SO (3) itself, which in turn is homeomorphic to theprojective space RP .Proof. Suppose that the first two points P and P of the given system arenot collinear with the origin O of R . Recall that we denote M the pointof R n corresponding to the system { P j } nj =1 and denote by P M the map P M : SO (3) → R n , ρ ρ · M . First let us show that P M is an immersion. It follows from the fact thatthe map P M commutes with the action of SO (3) (i.e., for any ρ, σ ∈ SO (3)we have P M ( ρσ ) = ρP M ( σ )) and with rescaling ( P λM = λP M ) that itis enough to check the differential of P M at the unity 1 ∈ SO (3) and for M = (1 , , , a, b, , . . . ), b = 0. Let us use the parametrization of SO (3) byEuler angles: ρ ( α, β, γ ) = cos α cos γ − cos β sin α sin γ − cos γ sin α − cos α cos β sin γ sin β sin γ cos β cos γ sin α − cos α cos γ cos α cos β cos γ − sin α sin γ − cos γ sin β sin α sin β cos α sin β cos β ! . The unity 1 ∈ SO (3) corresponds to α = β = γ = 0. A routine calculationreveals that the (transpose of) the Jacobi matrix of P M at 1 is J T = − b a . . . b . . . − − b − a . . . . The minor of J formed by the second, the fourth, and the sixth column is2 b = 0, thus P M is indeed an immersion.Note that the two non-collinear position vectors of the points P and P determine by the cross product the third vector which together with the firsttwo forms a basis of R . Since two different rotations ρ, σ ∈ SO (3) cannotact identically on the same basis, it follows that ρ · M = σ · M , and hence P M is injective. (cid:3) We now turn to the proof of Theorem 1.1.
Proof of Theorem 1.1 . Recall that the group SU (2) is a 2-fold covering ofthe group SO (3). We shall need explicit formulae for this covering. Thegroup SU (2) consists of the matrices ρ = (cid:18) α β − ¯ β ¯ α (cid:19) LIMINATION OF PARASITIC SOLUTIONS 9 where α, β ∈ C | α | + | β | = 1. If we represent a point P j = ( x ◦ j , y ◦ j , z ◦ j ) ∈ R by the matrix H j = (cid:18) z ◦ j x ◦ j + iy ◦ j x ◦ j − iy ◦ j − z ◦ j (cid:19) , then the action of the 3-dimensional rotation corresponding to the matrix ρ on the point P j can be computed as H j ρH j ρ − (see [5, Chapter 7 § ρ the projection of P j to the plane Oxy (represented as a complex number)is w j = c j α − ¯ c j β − z ◦ j αβ, where c j = x ◦ j + iy ◦ j . In particular, the projection of P j to the plane Oxy as a function of rotation ρ is a restriction of an analytic function (quadraticform) of variables α and β from the space C to the unit 3-dimensionalsphere S = { ( α, β ) ∈ C | | α | + | β | = 1 } ≃ SU (2) . Lemma 3.2.
The function F does not identically vanish on any of the orbitsof SO (3) on R n with the exception of the trivial case when the orbit reducesto the single point ∈ R n .Proof. Suppose that at least one of the points of the system { P j } nj =1 isdifferent from the origin. Without loss of generality we can assume that itis P . Case I : the points
O, P , . . . , P n lie on the same line. Then, the coordinatesof all the points P , . . . , P n are proportional to the coordinates of the point P , and the same holds for the projections of the points to the plane Oxy : c j = λ j c , λ j ∈ R , j = 2 , . . . , n. This relation is preserved under each rotation of the system around theorigin. Substituting w j = λ j w to the function F , we get F ( w , λ w , . . . , λ n w ) = w d F (1 , λ , . . . , λ n ) . Since the hypersurface X F = { F = 0 } has no real points, the last expressioncan vanish only if w = 0, i.e., if all points are disposed on the axis Oz . Case II : the points
O, P , . . . , P n don’t lie on the same line, but they arein the same plane. Without loss of generality we assume that the positionvectors of P and P are linearly independent and the rest are their linearcombinations. The same dependence holds also for the projections and ispreserved under any rotation of the system of points around the origin. Let c j = λ j c + µ j c , λ j , µ j ∈ R , j = 3 , . . . , n. The quadratic forms w = c α − ¯ c β − z ◦ αβ,w = c α − ¯ c β − z ◦ αβ are also linear independent, and thus define a 4-fold ramified covering of thecomplex 2-dimensional w -plane C by the ( α, β )-plane C . The formulae w = c α − ¯ c β − z ◦ αβ,w = c α − ¯ c β − z ◦ αβ,w j = λ j w + µ j w , j = 3 , . . . , n, define a map of the ( α, β )-plane C onto a 2-dimensional complex linearsubspace L of the space C n . If the function F vanished identically on theorbit of the system { P j } nj =1 , then it would vanish identically also on thesubspace L . Indeed, the set of zeros of an analytic function on C has realdimension 2 or 4 (or is empty, if the function is a non-zero constant), thussince our function vanishes on the 3-dimensional unit sphere S ⊂ C , itmust vanish on the all of C . Passing to the projectivization, vanishing of F on L would mean that the hypersurface X F ⊂ CP n − contained a line,which would contradict to our assumptions. Case III : the points
O, P , . . . , P n do not lie in one plane. As before, weassume from the beginning that the position vectors of the points P , P , P are linearly independent and c j = λ j c + µ j c + ν j c for some λ j , µ j , ν j ∈ R , j = 4 , . . . , n . Then, the quadratic forms w = c α − ¯ c β − z ◦ αβ,w = c α − ¯ c β − z ◦ αβ,w = c α − ¯ c β − z ◦ αβ are also linear independent and define a map of the ( α, β )-plane C ontoa non-degenerate (i.e., not splitting into 2 planes) quadratic cone in the3-dimensional complex w -space C . Note that this is nothing else but theaffine Veronese map from C to C . Formulae w = c α − ¯ c β − z ◦ αβ,w = c α − ¯ c β − z ◦ αβ,w = c α − ¯ c β − z ◦ αβ,w j = λ j w + µ j w + ν j w , j = 4 , . . . , n, define a map from the ( α, β )-plane C onto a 2-dimensional non-degeneratequadratic cone Q in the space C n . If the function b vanished identically onthe orbit of the system of points { P j } nj =1 , then it would vanish identicallyalso on the cone Q . But then the projectivization of Q (i.e., a non-degenerateconic) would be contained in the hypersurface X b ⊂ CP n − , which is againimpossible due to our assumptions. (cid:3) Let us continue the proof of Theorem 1.1. Consider the cases analogousto the proof of Lemma 3.2. If all the points of the system are concentratedat the origin, then the theorem is obvious. In
Case I we saw that both thefunctions Re F , Im F (and thus also the function F ) vanish if and only if allthe points lie on the coordinate axis Oz . Already this provides fixation of LIMINATION OF PARASITIC SOLUTIONS 11 the system. But for such position of points the function Im H vanishes aswell, so we have reduction too.In Cases II and III let us denote by ˜ F and ˜ H respectively the functionsobtained from F and H by the substitution(4) w j = c j α − ¯ c j β − z ◦ j αβ, j = 1 , . . . , n. They are polynomials of 2 complex variables α and β . The polynomial ˜ F is homogeneous of degree 4 d and, as we checked in Lemma 3.2, does notidentically vanish. It follows that it decomposes into linear factors. In otherwords, the set of zeros of ˜ F on C is a union of 4 d (counted with multi-plicities) complex 1-dimensional subspaces (lines). The map from C to C n defined by formulae (4) transforms such lines into 1-dimensional subspaces(lines) of the space C n . On the other hand, the function H does not identi-cally vanish on any of the complex lines passing through 0 ∈ C n of the space C n . Hence, also the function ˜ H does not identically vanish on any of thelines of the space C . Now Theorem 1.1 can be deduced from the followingsimple fact. Lemma 3.3.
Let f ( z ) be a non-constant complex polynomial considered asa function f : C → C . Assume also that f (0) = 0 . Then the set of pointsof the unit circle S = { z ∈ C | | z | = 1 } at which f takes real values is notempty and finite.Proof. The proof follows from standard theorems of analysis, but we provideit for the sake of completeness. If the polynomial f took no real values onthe unit circle, its imaginary part Im f would be either strictly positive orstrictly negative on the circle. But Im f = 0 at z = 0. It follows that theharmonic function Im f would attain its minimum (or maximum) inside theopen unit disc, but that would violate the maximum principle for Im f .Now assume that Im f vanishes on an infinite set of points of the unitcircle. Both the unit circle S and the zero set of Im f are real algebraicsets (because f is a polynomial) and, moreover, the circle is irreducible andremains irreducible after complexification. It follows that S ⊆ (Im f ) − (0),i.e., the harmonic polynomial Im f vanishes everywhere on the circle S .Again by the maximum principle for harmonic functions we can claim thatIm f = 0 on the whole unit disc and, since Im f is a polynomial, on thewhole complex plane C . In other words, the complex polynomial f takesonly real values. But since f is non-constant, this is a contradiction with,say, openness of an analytic map. (cid:3) To finish the proof of Theorem 1.1, apply Lemma 3.3 to the restrictionsof the polynomial ˜ H to each of the lines that constitute the set of zeros ofthe polynomial ˜ F . (cid:3) Remark . It is clear from the proof of Theorem 1.1 that the choice f =Im H can be changed to f = Re H . It is clear also that if one is interestedin reduction and fixation of non-degenerate system of points (in the sensethat O, P , . . . , P n are not contained in a plane), then the conditions aboutno lines and no real points on X F can be dropped, and a condition onlyneeded is that X F has no conics. In particular, there is no need for X F to be of even degree. Furthermore, also the condition about the conics can be somewhat weakened. A closer look at the projective conic definedby parametric equations (4) reveals that it has no real points. Thus, thesufficient condition on F is that X F has no conics with real points . Example . A system { P } with one point can be reduced (by SO (3))and fixed by only two equations x = y = 0. Let n = 2. A system withtwo points is always degenerate. The conditions about lines and conics inTheorem 1.1 become empty, so we can set F = w + w , G = w + w . But amuch simpler method not relying on Theorem 1.1 is to impose the equations z = z = 0, thus embedding the system to the coordinate plane Oxy , andadd one more equation ( x y ) + x y = 0 with Astrelin function, reducingto the plane case. Example . The first non-trivial case is that of n = 3 points. The hyper-surface X F in this case is a curve, so it suffices to ensure that it is not aconic, is irreducible and has no real points. The choice F = w + w + w defining the Fermat quartic and H = w + w + w works well. We can deal with the case of n = 4 points if the group G isextended to the full affine group of isometries of R . First we impose theconditions (2) fixing the center of mass. Then, only three of the pointsremain independent, and the situation essentially reduces to the case n = 3.We again can choose F = w + w + w + w to be the Fermat quartic. A straightforward verification shows that thesection of X F ⊂ CP by the plane w + w + w + w = 0is a non-singular (and thus irreducible) plane quartic with no real points.Thus this F and, say, H = w + w + w + w provide reduction and fixation of a system with 4 points with respect to thefull group of affine isometries of R . Example . It is still possible to write explicit function F for reductionand fixation of a system of 4 points with respect to the group SO (3) only.In [6, Theorem 3.1], van Luijk describes a family of quartics in CP definedover Q and of Picard number 1. The last condition means, in particular,that each of the surfaces has no lines and no conics. The equation of such aquartic X h is(5) w f + 2 w f = 3 g g + 6 h, where w , . . . , w are the homogeneous coordinates on CP , f , f , g , g are some explicitly given homogeneous polynomials whose precise form isnot important here, f i of degree 3, g j of degree 2, and h is any homogeneouspolynomial of degree 4. It follows that we can choose h to be the Fermat sumof 4-th degrees of w , . . . , w taken with a very big integral coefficient so that LIMINATION OF PARASITIC SOLUTIONS 13 the resulting quartic X h has no real points. Thus the function F definingthis quartic and H from Example 3.6 provide reduction and fixation of anysystem of 4 points in space. In fact, a version of the trick from Example 3.6applied to X h allows to deal also with systems of 5 points and with thegroup G the full group of affine isometries. For this, we take F ( w , . . . , w ) = w f + 2 w f − g g − h with h = − N ( w + · · · + w + w ). In view of the equations fixing thecenter of mass, the problem amounts to study of the quartic in CP withthe equation w f + 2 w f − g g + 6 N ( w + · · · + w + ( w + · · · + w ) ) = 0 . It has the same form as van Luijk’s equation (5), thus defines a quartic withno lines and conics, and for sufficiently big N it has no real points. Remark . In fact, Lemma 3.3 gives a method for constructing functionsproviding rotational reduction and fixation of systems of points in the planedifferent from Astrelin’s function A . Indeed, let us represent each point P j = ( x j , y j ) of a system { P j } nj =1 ⊂ R by the complex number w j = x j + iy j .Let H be any polynomial on C n such that H (0 , . . . ,
0) = 0 but H does notidentically vanish on any of the 1-dimensional vector subspaces of C n . Anexample of such H is given in Theorem 1.1. Then each system { P j } nj =1 inthe plane admits a reduction by SO (2) with respect to the function Im H and is fixed by this function with respect to SO (2). Example . Let { P j } nj =1 be the set of vertices of a regular polygon in R ( C ), and the initial positions of the points P j are the n th complex roots ofunity: P j = e pi ( j − /n , j = 1 , . . . , n. Let us choose the function H this time to be H = w n + w n + · · · + w n n . The action of SO (2) can now be represented as multiplication of each w j bya complex number e ϕi . The reader can easily calculate that the restrictionof Im H to the SO (2)-orbit of the system { P j } nj =1 isIm n X j =1 (cid:0) e πi ( j − /n e iϕ (cid:1) jn = Im (cid:16)X e ijnϕ (cid:17) = n X j =1 sin jψ, where ψ = nϕ . The last sum is 0 if ψ = 2 πk , and, if ψ = 2 πk , it can beexpressed as sin nψ sin ( n +1) ψ sin ψ . It follows that the regular polygon is fixed by Im H at its initial positionand at positions that differ from the initial one by a rotation by an angle ϕ = πkn , k = 1 , . . . , n −
1, or ϕ = πkn ( n +1) , k = 1 , . . . , n + n − H is not homogeneous has aconsequence that the rotation providing reduction is not invariant underscaling, i.e., multiplication of all the coordinates of all the points of a system by the same number. This should seem very unnatural from physicist’s pointof view. As our last remark we shall show that it is possible to replace H bya homogeneous function, but at the price of making the construction evenless explicit. Theorem 3.10.
Let F be a complex homogeneous polynomial satisfying theconditions of Theorem 1.1, and g ( x , y , . . . , x n , y n ) be a real homogeneouspolynomial of odd degree such that its complexification g C : C n → C definesa hypersurface X g C ⊂ CP n − which has no lines . Then each system of n points in R admits a reduction by SO (3) with respect to the functions f = Re F, f = Im F, f = g, considered as functions on R n , and at the same time is fixed by f , f , f with respect to SO (3) .Proof. The proof starts with the same argument as the proof of Theorem 1.1till we highlight several complex lines in C n , and the question reduces toproving that the restriction of g to each of these lines (considered now as realplanes in R n ) has a finite number of zeros on the unit circle. But indeed,the condition that we imposed on g ensures that g is not identically zero onany such plane L . On the other hand, the restriction g | L is a homogeneousfunction of odd degree. Thus g | L vanishes along several real lines, and theselines intersect the unit circle in a finite set of points. (cid:3) References [1] H. Clemens,
Curves on generic hypersurfaces , Ann. scient. ´Ec. Norm. Sup., 4 e s´erie (4) (1986), 629–636.[2] R. Connelly, The Rigidity of Certain Cabled Frameworks and the Second-Order Rigid-ity of Arbitrarily Triangulated Convex Surfaces , Advances in Math. (1980), 272–299.[3] K. Furukawa, Rational curves on hypersurfaces , J. Reine Angew. Math. (2012),157–188.[4] M. Gromov, Partial differential relations, Ergeb. der Math. 3. Folge, Springer Verlag,1986.[5] A. I. Kostrikin, Introduction to Algebra, Univesitext series, Springer Verlag, 1982.[6] R. van Luijk,
K3 surfaces with Picard number one and infinitely many rational points ,Algebra Number Theory (1) (2007), 1–15.[7] A. I. Markushevich, Short-course theory of analytic functions (in Russian), Mir, 2009.[8] R. Montgomery, The Three-body Problem and the Shape Sphere , American Math.Monthly (4) (2015), 299–321.[9] I. Kh. Sabitov,
Generalized Heron-Tartaglia’s formula and some of its consequences (in Russian), Matem. Sbornik (10) (1998), 105–134 (English translation inSbornik Mathematica. London Math. Soc. - 189:10.- p. 1533-1561).[10] I. Kh. Sabitov,
The volume as a metric invariant of polyhedra , Discrete and Compu-tatonal Geometry (4) (1998), 405–425.[11] I. Kh. Sabitov, Algebraic methods for solution of polyhedra , Russian Math. Surveys :(3) (2011), 445–505. Faculty of Mechanics and Mathematics, Lomonosov Moscow State Univer-sity, Leninskie Gory, Moscow GSP-1,119234, Russia
E-mail address : [email protected] The Department of Mathematical Modelling, Bauman Moscow State Tech-nical University, 2-ya Baumanskaya ul. 5, Moscow 105005, Russia
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