Gift Wrapping for Pretropisms

Transcription

1 Gift Wrapping for Pretropisms Jan Verschelde University of Illinois at Chicago Department of Mathematics, Statistics, and Computer Science jan Graduate Computational Algebraic Geometry Seminar Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

2 Outline 1 Pretropisms and Solution Sets an illustrative example pretropisms and tropisms 2 Applying Gift Wrapping computing cones of pretropisms sketching an algorithm in pseudocode running a test program in PHCpack Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

3 an illustrative example (x 2 x1 2)(x x x 3 2 1)(x 1 0.5) = 0 f(x 1, x 2, x 3 ) = (x 3 x1 3 )(x x x 3 2 1)(x 2 0.5) = 0 (x 2 x1 2)(x 3 x1 3)(x x x 3 2 1)(x 3 0.5) = 0 f 1 (0) = Z = Z 2 Z 1 Z 0 = {Z 21 } {Z 11 Z 12 Z 13 Z 14 } {Z 01 } 1 Z 21 is the sphere x x x = 0, 2 Z 11 is the line (x 1 = 0.5, x 3 = ), 3 Z 12 is the line (x 1 = 0.5, x 2 = 0.5), 4 Z 13 is the line (x 1 = 0.5, x 2 = 0.5), 5 Z 14 is the twisted cubic (x 2 x 2 1 = 0, x 3 x 3 1 = 0), 6 Z 01 is the point (x 1 = 0.5, x 2 = 0.5, x 3 = 0.5). Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

4 numerical irreducible decomposition Used in two papers in numerical algebraic geometry: first cascade of homotopies: 197 paths A.J. Sommese, J. Verschelde, and C.W. Wampler: Numerical decomposition of the solution sets of polynomial systems into irreducible components. SIAM J. Numer. Anal. 38(6): , equation-by-equation solver: 13 paths A.J. Sommese, J. Verschelde, and C.W. Wampler: Solving polynomial systems equation by equation. In Algorithms in Algebraic Geometry, Volume 146 of The IMA Volumes in Mathematics and Its Applications, pages , Springer-Verlag, The mixed volume of the Newton polytopes of this system is 124. By theorem A of Bernshteǐn, the mixed volume is an upper bound on the number of isolated solutions. Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

5 three Newton polytopes (x 2 x1 2)(x x x 3 2 1)(x 1 0.5) = 0 f(x 1, x 2, x 3 ) = (x 3 x1 3 )(x x x 3 2 1)(x 2 0.5) = 0 (x 2 x1 2)(x 3 x1 3)(x x x 3 2 1)(x 3 0.5) = 0 Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

6 looking for solution curves The twisted cubic is (x 1 = t, x 2 = t 2, x 3 = t 3 ). We look for solutions of the form x 1 = t v 1, v 1 > 0, x 2 = c 2 t v 2, c 2 C, x 3 = c 3 t v 3, c 3 C. Substitute x 1 = t, x 2 = c 2 t 2, x 3 = c 3 t 3 into f (0.5c 2 0.5)t 2 + O(t 3 ) = 0 f(x 1 = t, x 2 = c 2 t 2, x 3 = c 3 t 3 ) = (0.5c 3 0.5)t 3 + O(t 5 ) = 0 0.5(c 2 1.0)(c 3 1.0)t 5 + O(t 7 ) conditions on c 2 and c 3. How to find (v 1, v 2, v 3 ) = (1, 2, 3)? Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

7 faces of Newton polytopes Looking at the Newton polytopes in the direction v = (1, 2, 3): Selecting those monomials supported on the faces 0.5x 2 0.5x1 2 = 0 v f(x 1, x 2, x 3 ) = 0.5x 3 0.5x1 3 = 0 0.5x 2 x x 3x x 3x x1 5 = 0 Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

8 degenerating the sphere (x 2 x1 2)(x x x 3 2 1)(x 1 0.5) = 0 f(x 1, x 2, x 3 ) = (x 3 x1 3 )(x x x 3 2 1)(x 2 0.5) = 0 (x 2 x1 2)(x 3 x1 3)(x x x 3 2 1)(x 3 0.5) = 0 As x 1 = t 0: x 2 (x2 2 + x 3 2 1)( 0.5) = 0 (1,0,0) f(x 1, x 2, x 3 ) x 3 (x2 2 + x 3 2 1)(x 2 0.5) = 0 x 2 x 3 (x2 2 + x 3 2 1)(x 3 0.5) = 0 As x 2 = s 0: x 1 2(x x 3 2 1)(x 1 0.5) = 0 (0,1,0) f(x 1, x 2, x 3 ) (x 3 x1 3 )(x x 3 2 1)( 0.5) = 0 x1 2(x 3 x1 3)(x x 3 2 1)(x 3 0.5) = 0 Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

9 more faces of Newton polytopes Looking at the Newton polytopes along v = (1,0,0) and v = (0, 1, 0): (1,0,0) f(x 1, x 2, x 3 ) = x 2 (x2 2 + x 3 2 1)( 0.5) x 3 (x2 2 + x 3 2 1)(x 2 0.5) x 2 x 3 (x2 2 + x 3 2 1)(x 3 0.5) (0,1,0) f(x 1, x 2, x 3 ) = x1 2(x x 3 2 1)(x 1 0.5) (x 3 x1 3 )(x x 3 2 1)( 0.5) x1 2(x 3 x1 3)(x x 3 2 1)(x 3 0.5) Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

10 faces of faces The sphere degenerates to circles at the coordinate planes. (1,0,0) f(x 1, x 2, x 3 ) = x 2 (x2 2 + x 3 2 1)( 0.5) x 3 (x2 2 + x 3 2 1)(x 2 0.5) x 2 x 3 (x2 2 + x 3 2 1)(x 3 0.5) (0,1,0) f(x 1, x 2, x 3 ) = x1 2(x x 3 2 1)(x 1 0.5) (x 3 x1 3 )(x x 3 2 1)( 0.5) x1 2(x 3 x1 3)(x x 3 2 1)(x 3 0.5) Degenerating even more: x 2 (x3 2 1)( 0.5) (0,1,0) (1,0,0) f(x 1, x 2, x 3 ) = x 3 (x3 2 1)( 0.5) x 2 x 3 (x3 2 1)(x 3 0.5) The factor x is shared with (1,0,0) (0,1,0) f(x 1, x 2, x 3 ). Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

11 representing a solution surface The sphere is two dimensional, x 1 and x 2 are free: x 1 = t 1 x 2 = t 2 x 3 = 1 + c 1 t1 2 + c 2t2 2. For t 1 = 0 and t 2 = 0, x 3 = 1 is a solution of x 3 1 = 0. Substituting (x 1 = t 1, x 2 = t 2, x 3 = 1 + c 1 t c 2t 2 2 ) into the original system gives linear conditions on the coefficients of the second term: c 1 = 0.5 and c 2 = 0.5. Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

12 Outline 1 Pretropisms and Solution Sets an illustrative example pretropisms and tropisms 2 Applying Gift Wrapping computing cones of pretropisms sketching an algorithm in pseudocode running a test program in PHCpack Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

13 notations Given p, q C[x ±1, y ±1, z ±1 ], do p and q have a common factor? For example: { p = (x 2 + y 2 + z 2 1)(y x 2 ) q = (x 2 + y 2 + z 2 1)(z x 3 ) In our polyhedral approach, we write p and q as p(x, y, z) = a A c a x a 1 y a 2 z a 3 and q(x, y, z) = b A c b x b 1 y b 2 z b 3 where A = { a Z 3 c a 0 } and B = { b Z 3 c b 0 } are the support sets respectively of p and q. A spans the Newton polytope P = conv(a), B spans Q = conv(b). Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

14 inner normals and initial forms Denote by, the inner product: a, b = a 1 b 1 + a 2 b 2 + a 3 b 3. For v 0, the face in v P of P = conv(a) is spanned by in v A with in v A = { a A a, v = min b, v }. b A We use in v A because of initial forms of polynomials: in v p(x, y, z) = a in va c a x a1 y a2 z a3 where p(x, y, z) = a A We may take v, with integer coordinates v 1, v 2, and v 3, normalized so that gcd(v 1, v 2, v 3 ) = 1. This normalization gives unique normals to all proper facets. c a x a 1 y a 2 z a 3. Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

15 pretropisms Let (A, B) be two supports. A pretropism for (A, B) is a vector v 0: #in v A 2 and #in v B 2. Denote by T(A, B) the set { v 0 #in v A 2 and #in v B 2 }. A pretropism is a candidate for a tropism. A tropism v is a pretropism for which a root of the initial form system in v f(x) = 0 determines the leading coefficients of a Puiseux series expansion of a solution component of f(x) = 0. Proposition If T(A, B) =, then for any two Laurent polynomials p and q with respective support sets A and B, p and q have no common factor. Proof. Suppose p and q have a nontrivial common factor f, i.e.: p = p 1 f and q = q 1 f. Denote the Newton polytope of f by F, then P = P 1 + F and Q = Q 1 + F, where P, P 1, Q, and Q 1 are the respective Newton polytopes of p, p 1, q, and q 1. All normals to faces of F are tropisms. Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

16 predicting a common factor Recall the example: { p = (x 2 + y 2 + z 2 1)(y x 2 ) q = (x 2 + y 2 + z 2 1)(z x 3 ) The set {(1, 0, 0), (0, 1, 0), (0, 0, 1), ( 1, 1, 1)} contains all normalized vectors to the facets of the simplex, the Newton polytope of the common factor of p and q. Among the curves common to p and q we get the equations of the twisted cubic via the initial forms in ( 1, 2, 3) p = z 2 (y x 2 ) and in (+1,+2,+3) p = 1(y x 2 ). Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

17 classifying pretropisms Redundant tropisms: for curves restrict to first positive component. For surfaces: every edge of the Newton polytope of a common factor will also be a pretropism. For two Newton polytopes P and Q, we classify pretropisms as follows: A facet pretropism is an inner normal to a facet common to both P and Q. We call this common facet a tropical prefacet. An edge pretropism is a pretropism not contained in any tropical prefacet. Edges of P and Q that are parallel to each other are always contained in a larger face tuple (in v P, in v Q) for some pretropism v. Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

18 Outline 1 Pretropisms and Solution Sets an illustrative example pretropisms and tropisms 2 Applying Gift Wrapping computing cones of pretropisms sketching an algorithm in pseudocode running a test program in PHCpack Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

19 applying gift wrapping Recall the geometric intuition of gift wrapping: view a supporting hyperplane as wrapping paper, the paper first touches a vertex, then an edge, etc. planes supporting facets are rotated along ridges. Consider as given a tuple of Newton polytopes, the pretropisms correspond to those facets of the Minkowski sum of the Newton polytopes, that are spanned by sums of edges of the polytopes. On the complexity: Storing the entire face lattice of a Newton polytope of a sparse polynomial has an acceptable complexity. Storing the entire face lattice of the sum of the Newton polytopes is not efficient and most likely not even desirable. Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

20 two Newton polytopes in 3-space: facet pretropisms Given are two support sets A and B, A Z 3 n A and B Z 3 n B. Could the polynomials p (supported on A) and q (supported on B) have a common factor? This question is equivalent for two facets: to share the same inner normal. F A of conv(a) and F B of conv(b) If the components of each inner normal vector to a facet are normalized so their greatest common divisor equals one, then the problem of finding a facet pretropism is reduced to sorting: 1 Sort the inner normals of the facets to A and B lexicographically. 2 Merge the sorted lists of inner normals. Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

21 two Newton polytopes in 3-space: edge pretropisms Given are two support sets A and B, A Z 3 n A and B Z 3 n B. The search for a pretropism starts at a facet F A of conv(a): The vertex points that span the facet are ordered: two consecutive vertex points of the facet span an edge e A. Two neighboring facets are connected through exactly one edge: the facet normal v of F A and the normal u to the neighboring facet span the inner normal cone of the connecting edge e A. With e A and its cone spanned by {u, v}, we explore B: For random real λ u,λ v > 0, let w = λ u u + λ v v, then in w B = {b}. Run over all edges e B incident to b and check the pair (e A, e B ): if the intersection of the inner normal cones to e A and e B is not empty, then their intersection contains a pretropism. Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

22 Outline 1 Pretropisms and Solution Sets an illustrative example pretropisms and tropisms 2 Applying Gift Wrapping computing cones of pretropisms sketching an algorithm in pseudocode running a test program in PHCpack Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

23 sketching an algorithm in pseudocode Input: (A, B), A Z 3 n A, B Z 3 n B. Output: T f (A, B); T e (A, B). 1. F A := conv(a); F B := conv(b); 2. T f (A, B) := { v v is normal to f F A F B }; 3. for all v normal to facet f F A, v T f (A, B) do 3.1 let e A be edge of A, not visited before; 3.2 let u: e A = in u A in v A; 3.3 let b be a vertex of in u+v B; 3.4 for all edges e B incident to b do if w (e A, e B ) is edge pretropism then T e (A, B) := T e (A, B) {w}; set b to unvisited vertex of B; goto 3.4; Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

24 a crude cost analysis Assuming a uniform cost of arithmetic ( multiprecision): The cost of computing all facet pretropisms: Let N A = #facets of conv(a) and N B = #facets of conv(b), set M = max(n A, N B ), then the cost reduces to sorting, which is O(M log 2 (M)). The cost of computing all edge pretropisms: Let N A = #edges of conv(a) and N B = #edges of conv(b), set M = max(n A, N B ), then the cost reduces to making combinations: O(M 2 ). For a sharper bound, let m be the maximum number of edges per facet, then the cost for all edge pretropisms becomes O(Mm). Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

25 Outline 1 Pretropisms and Solution Sets an illustrative example pretropisms and tropisms 2 Applying Gift Wrapping computing cones of pretropisms sketching an algorithm in pseudocode running a test program in PHCpack Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

26 running a test program in PHCpack The code for pretropisms is not (yet) wrapped to phcpy. Running the test program ts_pretrop on the illustrative example: The list of facet pretropisms : which corresponds to the inner normals of a simplex, the Newton polytope of the common factor x x x Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27

27 pretropisms for the space curves Take the first two polynomials of the illustrative example: p : (y-x**2)*(x**2 + y**2 + z**2-1)*(x - 0.5); q : (z-x**3)*(x**2 + y**2 + z**2-1)*(y - 0.5); The output of the test program ts_pretrop contains The edge tropisms via giftwrapping : We recognize the twisted cubic (t 1, t 2, t 3 ). Jan Verschelde (UIC) gift wrapping for pretropisms 11 September / 27