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Exact and Approximate Algorithms for Resultant Polytopes

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Exact and Approximate Algorithms for Resultant Polytopes EuroCG 2012, Assisi, Italy, March 19{21, 2012 Exact and approximate algorithms for resultant polytopes Ioannis Z. Emiris∗ Vissarion Fisikopoulos∗ Christos Konaxisy n Abstract where e0; : : : ; en form an affine basis of R : e0 is the zero vector, ei = (0;:::; 0; 1; 0;:::; 0); i = 1; : : : ; n. P We develop an incremental algorithm to compute the Clearly, jAj = i jAij, where j · j denotes cardinality. Newton polytope of the resultant, aka resultant poly- By Cayley's trick ([13, sec.5]) the regular tight mixed tope, or its projection along a given direction. Our subdivisions of the Minkowski sum A0 + ··· + An are algorithm exactly computes vertex- and halfspace- in bijection with the regular triangulations of A; the representations of the resultant polytope using an or- latter correspond to the vertices of the secondary poly- acle producing resultant vertices in a given direction. tope Σ(A). It is output-sensitive as it uses one oracle call per The Newton polytope of a polynomial is the convex vertex. We implement our algorithm using the exper- hull of the set of exponent vectors of monomials with imental CGAL package triangulation. A variant of nonzero coefficient. Given n+1 polynomials in n vari- the algorithm computes successively tighter inner and ables, with fixed exponent sets Ai, i = 0; : : : ; n, and outer approximations: when these polytopes have, re- symbolic coefficients, their sparse (or toric) resultant spectively, 90% and 105% of the true volume, runtime R is a polynomial in these coefficients which vanishes is reduced up to 25 times. Compared to tropical ge- exactly when the polynomials have a common root. ometry software, ours is faster up to dimensions 5 or The resultant is the most fundamental tool in elimina- 6, and competitive in higher dimensions. Compared tion theory, and is instrumental in system solving; it to lrs, cdd, and polymake, the computation of con- is also important in changing representation of para- vex hull is fastest along with polymake. The resul- metric hypersurfaces. Our algorithms compute the tant is fundamental in algebraic elimination and in Newton polytope of the resultant N(R), or resultant implicitizing parametric hypersurfaces: we compute polytope, in particular when some of the coefficients the Newton polytope of surface equations in < 1sec, are not symbolic, in which case we seek a projection when there are < 100 terms in the parametric poly- of the resultant polytope. nomials, which includes all common instances in geo- We exploit an equivalence relation defined on the metric modeling. Our method computes instances of secondary vertices. The class representatives cor- 5, 6 or 7-dimensional polytopes with 35K, 23K or 500 respond bijectively to the resultant vertices. This vertices, respectively, in < 2hr. information defines an oracle producing a resultant Keywords. general dimension, convex hull, regu- vertex in a given direction, and avoids computing lar triangulation, polynomial resultant, CGAL imple- Σ(A), which has much more vertices than N(R). Al- mentation, experimental complexity though there exist efficient software computing Σ(A) [12], it is useless in computing resultant polytopes. For instance, in implicitizing parametric surfaces with 1 Introduction < 100 input terms, we compute the Newton poly- tope of the equations in < 1sec, whereas Σ(A) is in- n Given pointsets A0;:::;An ⊂ Z , we define the Cay- tractable. ley pointset Moreover we compute some orthogonal projection m n of N(R), denoted Π, in R : [ A := (A × fe g) ⊂ 2n; e 2 n; (1) i i Z i N π : jAj ! m : N(R) ! Π; m ≤ jAj: i=0 R R ∗Department of Informatics and Telecommunications, Na- By reindexing, this is the subspace of the first m tional and Kapodistrian University of Athens, Athens, Greece. coordinates. It is possible that none of the coeffi- femiris,[email protected]. Partial support from project cients is specialized, hence m = jAj, π is trivial, and \Computational Geometric Learning", which acknowledges the Π = N(R). Assuming the specialized coefficients financial support of the Future and Emerging Technologies (FET) programme within the FP7 for Research of the Euro- take sufficiently generic values, Π is the Newton poly- pean Commission, under FET-Open grant number: 255827. tope of the corresponding specialization of R. yArchimedes Center for Modeling, Analysis & Computa- Let us review previous work. Sparse (or toric) elim- tion (ACMAC), University of Crete, Heraklio, Greece, ckon- ination theory was introduced in [9], where N(R) is [email protected]. Enjoys support from the FP7-REGPOT- 2009-1 project \Archimedes Center for Modeling, Analysis and described via Cayley's trick. In [13, sec.6] is proven Computation" that N(R) is 1-dimensional iff jAij = 2, for all i, the 28th European Workshop on Computational Geometry, 2012 only planar polytope is the triangle, whereas the only Algorithm 1: ComputeΠ (A0;:::;An; π) 3-dimensional ones are the tetrahedron, the square- n jAj m based pyramid, and the polytope of two univariate tri- Input : A0;:::;An ⊂ Z , π : R ! R , H nomials. Following [13, Thm.6.2], the 4-dimensional H-, V-rep. Q ;Q, triang. TQ of Q ⊆ Π H polytopes include the 4-simplex, some N(R) obtained Output: H-, V-rep. Q ;Q, triang. TQ of Q = Π by pairs of polynomials, and those of 3 trinomials, Sn which we can investigate with our code: for example, A 0 (Ai × ei); Hillegal ; H we computed one with f-vector (19; 57; 57; 19). Trop- foreach H 2 Q do Hillegal Hillegal [ fHg ical geometry is a polyhedral analogue of algebraic while Hillegal 6= ; do geometry which gives alternative ways of recovering select H 2 Hillegal; Hillegal Hillegal n fHg resultant polytopes [10]. w is the outer normal vector of H For a more detailed presentation of the background v ExtremeΠ(A; w; π) of this paper see [5]. if v2 = H \ Q then H Qtemp conv(Q [ fvg) 2 Algorithms and complexity foreach (d − 1)-face f 2 TQ, f ⊂ @H do TQ TQ [ ffaces of conv(f; v)g 0 H H This section presents our exact and approximate algo- foreach H 2 fQ n Qtempg do 0 rithms for computing an orthogonal projection Π of Hillegal Hillegal n fH g N(R) without computing N(R), and analyzes their 0 H H foreach H 2 fQtemp n Q g do asymptotic complexity. 0 Hillegal Hillegal [ fH g Given pointset V , reg subdivision(V; w) computes H H the regular subdivision of its convex hull by project- Q Q [ fvg; Q Qtemp ing the upper hull of V lifted by the linear functional H return Q; Q ;TQ w, and conv(V ) computes the H-representation of its convex hull. ExtremeΠ(A; w; π) computes a point −1 in Π, extremal in the direction w, by refining the the preimage π (f) ⊂ N(R) of the facet f of Q de- output of reg subdivision(A; w) into a regular trian- fined by H, is not a Minkowski summand of a face of Σ(A) having normal w. Otherwise, there are two gulation T of A, then returns π(ρT ), where ρT is the b resultant vertex corresponding to T . It is clear that, cases: either v 2 H and v 2 Q, thus the algorithm triangulation T constructed by ExtremeΠ, is regular simply decides hyperplane H is legal, or v 2 H and v2 = Q, in which case the algorithm again decides H and corresponds to some vertex φT of Σ(A) which ~ jAj × is legal but also insert v to Q. maximizes inner product with wb = (w; 0) 2 (R ) . The initialization algorithm computes an inner ap- Let us examine degenerate cases that may appear proximation of Π in both V- and H-representations during execution of Alg. 1. Let w be a normal to (denoted Q; QH , resp.), and triangulated. First, it a supporting hyperplane H of a facet of Q s.t. the face f of N(R) extremal wrt w contains a vertex ρ calls ExtremeΠ(A; w; π) for w 2 W ⊂ (Rm)×; set W b T is either random or contains, say, vectors in the 2m which projects to relint(π(f)), where relint(·) denotes coordinate directions. Then, it updates Q by adding relative interior. Then, if ExtremeΠ(A; w; π) returns ExtremeΠ(A; w; π) and ExtremeΠ(A; −w; π), where π(ρT ), this is a point on @Π but not a vertex of Π. Of course, in subsequent steps of the algorithm, the ver- w is normal to hyperplane H ⊂ Rm containing Q, as long as either of these points lies outside H. We stop tices of π(f) will be computed, but this jeopardizes when these points do no longer increase dim(Q). the output-sensitivity of the algorithm. We resolve such degeneracies by adding an infinitesimal generic perturbation vector to w, thus obtaining w . Since Lemma 1 The initialization algorithm computes p the perturbation is arbitrarily small, wcp shall be nor- Q ⊆ Π s.t. dim(Q) = dim(Π). mal to a vertex of f extremal wrt w but projecting Incremental Alg. 1 computes both V- and H- to a vertex of π(f). The perturbation can be im- representations of Π and a triangulation of Π, given plemented in ExtremeΠ, without affecting any other an inner approximation Q; QH of Π computed at ini- parts of the algorithm. In practice, our implementa- tialization. A hyperplane H is legal if it is a sup- tion does avoid degenerate cases. porting hyperplane to a facet of Π, otherwise it is Lemma 2 Let vertex v be computed by illegal. At every step of Alg. 1, we compute v = ExtremeΠ(A; w; π), where w is normal to a support- ExtremeΠ(A; w; π) for a supporting hyperplane H of ing hyperplane H of Q, then v 62 H , H is not a a facet of Q with normal w.
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