on Newton polytopes, tropisms, and Puiseux series to solve polynomial systems

Transcription

1 on Newton polytopes, tropisms, and Puiseux series to solve polynomial systems Jan Verschelde joint work with Danko Adrovic University of Illinois at Chicago Department of Mathematics, Statistics, and Computer Science jan SIAM Conference on Discrete Mathematics Interactions between Computer Algebra and Discrete Mathematics June 2012, Dalhousie University, Halifax, NS, Canada Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

2 Outline 1 Introduction our problem: solving sparse polynomial systems Newton polytopes, Puiseux series, initial forms, and tropisms 2 Solving Binomial Systems a very sparse class of polynomial systems unimodular coordinate transformations solutions to the cyclic 4-roots system 3 Polyhedral Methods for Algebraic Sets computing pretropisms with the Cayley embedding Puiseux series for algebraic sets 4 Application to the Cyclic n-roots Problem a series solution for cyclic 8-roots curves an exact representation for cyclic 9-roots surfaces Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

3 problem statement A polynomial in n variables x = (x 0, x 1,..., x n 1 ) consists of a vector of nonzero complex coefficients with corresponding exponents in A: f(x) = a A c a x a, c a C\{0}, x a = x a 0 0 x a 1 1 x a n 1 n 1. Solve f(x) = 0, f = (f 0, f 1,..., f N 1 ) with supports (A 0, A 1,...,A N 1 ). Systems are sparse: few monomials have a nonzero coefficient. For a Z n, we consider Laurent polynomials, f C[x ±1 ] only solutions with coordinates in C = C\{0} matter. Many applications give rise to symmetric polynomial systems. The solution set is invariant under permutations of the variables. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

4 the cyclic n-roots system x 0 + x 1 + +x n 1 = 0 x 0 x 1 + x 1 x 2 + +x n 2 x n 1 + x n 1 x 0 = 0 n 1 j+i 1 f(x) = i = 3, 4,...,n 1 : x k mod n = 0 Lemma (Backelin) j=0 x 0 x 1 x 2 x n 1 1 = 0. If m 2 divides n, then the cyclic n-roots system has a solution set of dimension m 1. J. Backelin: Square multiples n give infinitely many cyclic n-roots. Reports, Matematiska Institutionen, Stockholms Universitet, k=j Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

5 Newton polytopes and Puiseux series The sparse structure is modeled by its Newton polytope. Definition Consider the support A Z n of f(x) = c a x a, c a C\{0}. a A The Newton polytope of f is the convex hull of A. Definition Consider a curve C defined by f(x) = 0. A Puiseux series of the curve C has the form { x0 = t v 0 x k = z k t v k(1+o(t)), k = 1, 2,...,n 1, where (z 1,...,z n 1 ) (C ) n 1. The Newton-Puiseux algorithm is in Walker s Algebraic Curves, Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

6 initial forms and tropisms Denote the inner product of vectors u and v as u, v. Definition Let v Z n \{0} be a direction vector. Consider f(x) = a A c a x a. The initial form of f in the direction v is in v (f) = c a x a, where m = min{ a, v a A }. a A a, v = m Definition Let the system f(x) = 0 define a curve. A tropism consists of the leading powers (v 0, v 1,..., v n 1 ) of a Puiseux series of the curve. The leading coefficients of the Puiseux series satisfy in v (f)(x) = 0. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

7 some references D.N. Bernshteǐn. The number of roots of a system of equations. Functional Anal. Appl., 9(3): , B. Sturmfels. Gröbner Bases and Convex Polytopes, AMS, T. Bogart, A.N. Jensen, D. Speyer, B. Sturmfels, and R.R. Thomas. Computing tropical varieties. J. Symbolic Computation, 42(1):54 73, A.N. Jensen, H. Markwig, and T. Markwig. An algorithm for lifting points in a tropical variety. Collectanea Mathematica, 59(2): , S. Payne. Fibers of tropicalization. Mathematische Zeitschrift, 262(2): , D. Adrovic and J. Verschelde. Tropical algebraic geometry in Maple: A preprocessing algorithm for finding common factors to multivariate polynomials with approximate coefficients. J. Symbolic Computation, 46(7): , Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

8 relevant software Maple, of course... cddlib by Komei Fukuda and Alain Prodon implements the double description method to efficiently enumerate all extreme rays of a general polyhedral cone. Gfan by Anders Jensen to compute Gröbner fans and tropical varieties uses cddlib. The Singular library tropical.lib by Anders Jensen, Hannah Markwig and Thomas Markwig for computations in tropical geometry. Macaulay2 interfaces to Gfan. Sage interfaces to Gfan. PHCpack (published as Algorithm 795 ACM TOMS) provides our numerical blackbox solver. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

9 binomial systems Definition A binomial system has exactly two monomials with nonzero coefficient in every equation. The binomial equation c a x a c b x b = 0, a, b Z n, c a, c b C\{0}, has normal representation x a b = c b /c a. A binomial system of N equations in n variables is then defined by an exponent matrix A Z N n and a coefficient vector c (C ) N : x A = c. Solution sets of binomial systems are related to toric varieties. Solution sets of binomial systems can be represented exactly by the first term of their Puiseux series. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

10 an example Consider as an example for x A = c the system { x 2 0 x 1 x 4 2 x = 0 x 0 x 1 x 2 x 3 1 = 0 A = [ ] T c = [ 1 1 ]. As basis of the null space of A we can for example take u = ( 3, 2, 1, 0) and v = ( 2, 1, 0, 1). The vectors u and v are tropisms for a two dimensional algebraic set. Placing u and v in the first two rows of a matrix M, extended so det(m) = 1, we obtain a coordinate transformation, x = y M : M = x 0 = y 3 0 y 2 1 y 2 x 1 = y 2 0 y 1y 3 x 2 = y 0 x 3 = y 1. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

11 monomial transformations By construction, as Au = 0 and Av = 0: MA = = = B. The corresponding monomial transformation x = y M performed on x A = c yields y MA = y B = c, eliminating the first two variables: { y 2 2 y 3 1 = 0 y 2 y 3 1 = 0. Solving this reduced system gives values z 2 and z 3 for y 2 and y 3. Leaving y 0 and y 1 as parameters t 0 and t 1 we find as solution (x 0 = z 2 t 3 0 t 2 1, x 1 = z 3 t 2 0 t 1, x 2 = t 0, x 3 = t 1 ). Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

12 unimodular coordinate transformations Definition A unimodular coordinate transformation x = y M is determined by an invertible matrix M Z n n : det(m) = ±1. For a d dimensional solution set of a binomial system: 1 The null space of A gives d tropisms, stored in the rows of a d-by-n-matrix B. 2 Compute the Smith normal form S of B: UBV = S. 3 There are three cases: 1 U = I M = V 1 2 If U I and S has ones on its diagonal, then extend U 1 with an identity matrix to form M. 3 Compute the Hermite normal form H of B and let D be the diagonal elements of H, then M = [ D 1 B 0 I ]. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

13 cyclic 4-roots and binomial systems x 0 + x 1 + x 2 + x 3 = 0 x f(x) = 0 x 1 + x 1 x 2 + x 2 x 3 + x 3 x 0 = 0 x 0 x 1 x 2 + x 1 x 2 x 3 + x 2 x 3 x 0 + x 3 x 0 x 1 = 0 x 0 x 1 x 2 x 3 1 = 0 Looking for a special solution, we apply the binomial system solver to x 0 + x 2 = 0 x 1 + x 3 = 0 x 0 x 1 + x 1 x 2 = 0 x 2 x 3 + x 3 x 0 = 0 x 0 x 1 x 2 + x 2 x 3 x 0 = 0 x 1 x 2 x 3 + x 3 x 0 x 1 = 0 x 0 x 1 x 2 x 3 1 = 0 Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

14 the output of phc -b 4 5 t1 - t1 + x0 - t1^1; x2 - ( E-16*i)*t1^1; x1 - (-1)*t1^-1; x3 - ( E-16*i)*t1^-1; 4 5 t1 - t1 + x0 - t1^1; x2 - ( E-16*i)*t1^1; x1 - ( E-16*i)*t1^-1; x3 - ( E-16*i)*t1^-1; This output corresponds to the two solutions (x 0 = t, x 1 = t 1, x 2 = t, x 3 = t 1 ) and (x 0 = t, x 1 = t 1, x 2 = t, x 3 = t 1 ) of the original problem. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

15 the Cayley embedding an example { p = (x0 x 2 1 )(x 0 + 1) = x x 0 x 2 1 x 0 x 2 1 = 0 q = (x 0 x 2 1 )(x 1 + 1) = x 0 x 1 + x 0 x 3 1 x 2 1 = 0 The Cayley polytope is the convex hull of {(2, 0, 0),(1, 0, 0), (1, 2, 0),(0, 2, 0)} {(1, 1, 1),(1, 0, 1), (0, 3, 1),(0, 2, 1)}. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

16 facet normals and initial forms The Cayley polytope has facets spanned by one edge of the Newton polygon of p and one edge of the Newton polygon of q. Consider v = (2, 1, 0). { ( in(2,1) (p) = in (2,1) x x 0 x1 2x 0 x1 2 ) = x0 x1 2 in (2,1) (q) = in (2,1) ( x0 x 1 + x 0 x 3 1 x 2 1) = x0 x 2 1 Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

17 computing all pretropisms Definition A nonzero vector v is a pretropism for the system f(x) = 0 if #in v (f k ) 2 for all k = 0, 1,...,N 1. Application of the Cayley embedding to (A 0, A 1,..., A N 1 ): N 1 E = { (a, 0) a A 0 } { (a, e k ) a A k } Z n+n 1, k=1 where 0, e 1 = (1, 0,...,0), e 2 = (0, 1,...,0),..., e N 1 = (0, 0,..., 1) span the standard unit simplex in R N 1. The set of all facet normals to the convex hull of E contains all normals to facets spanned by at least two points of each support. We used cddlib to compute all pretropisms of the cyclic n-roots system, up to n = 12. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

18 cones of pretropisms Definition A cone of pretropism is a polyhedral cone spanned by pretropisms. If we are looking for an algebraic set of dimension d and if there are no cones of vectors perpendicular to edges of the Newton polytopes of f(x) = 0 of dimension d, then the system f(x) = 0 has no solution set of dimension d that intersects the first d coordinate planes properly; otherwise if a d-dimensional cone of vectors perpendicular to edges of the Newton polytopes exists, then that cone defines a part of the tropical prevariety. For the cyclic 9-roots system, we found a two dimensional cone of pretropisms. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

19 solving the cyclic 4-roots system x 0 + x 1 + x 2 + x 3 = 0 x f(x) = 0 x 1 + x 1 x 2 + x 2 x 3 + x 3 x 0 = 0 x 0 x 1 x 2 + x 1 x 2 x 3 + x 2 x 3 x 0 + x 3 x 0 x 1 = 0 x 0 x 1 x 2 x 3 1 = 0 One tropism v = (+1, 1,+1, 1) with in v (f)(z) = 0: x 1 + x 3 = 0 x in v (f)(x) = 0 x 1 + x 1 x 2 + x 2 x 3 + x 3 x 0 = 0 x 1 x 2 x 3 + x 3 x 0 x 1 = 0 x 0 x 1 x 2 x 3 1 = 0 x 0 = y +1 0 x 1 = y 1 0 y 2 x 2 = y +1 0 y 3 x 3 = y 1 0 y 4 The system in v (f)(y) = 0 has two solutions. We find two solution curves: ( t, t 1, t, t 1) and ( t, t 1, t, t 1). Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

20 Puiseux series for algebraic sets Proposition If f(x) = 0 is in Noether position and defines a d-dimensional solution set in C n, intersecting the first d coordinate planes in regular isolated points, then there are d linearly independent tropisms v 0, v 1,...v d 1 Q n so that the initial form system in v0 (in v1 ( in vd 1 (f) ))(x = y M ) = 0 has a solution c (C\{0}) n d. This solution and the tropisms are the leading coefficients and powers of a generalized Puiseux series expansion for the algebraic set: x 0 = t v 0,0 0 x 1 = t v 0,1 0 t v 1,1 1. x d 1 = t v 0,d 1 0 t v 1,d 1 1 t v d 1,d 1 d 1 x d = c 0 t v 0,d 0 t v 1,d 1 t v d 1,d d 1 + x d+1 = c 1 t v 0,d+1 0 t v 1,d+1 1 t v d 1,d+1 d 1 +. x n = c n d 1 t v 0,n 1 0 t v 1,n 1 1 t v d 1,n 1 d 1 + Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

21 our polyhedral approach For every d-dimensional cone C of pretropisms: 1 We select d linearly independent generators to form the d-by-n matrix A and the unimodular transformation x = y M. 2 If in v0 (in v1 ( in vd 1 (f) ))(x = y M ) = 0 has no solution in (C ) n d, then return to step 1 with the next cone C, else continue. 3 If the leading term of the Puiseux series satisfies the entire system, then we report an explicit solution of the system and return to step 1 to process the next cone C. Otherwise, we take the current leading term to the next step. 4 If there is a second term in the Puiseux series, then we have computed an initial development for an algebraic set and report this development in the output. Note: to ensure the solution of the initial form system is not isolated, it suffices to compute a series development for a curve. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

22 applied to the cyclic 8-roots system Our approach applied to the cyclic 8-roots system: 831 facet normals (computed with cddlib) 29 pretropism generators 5 lead to initial forms with solutions (1, 1, 0, 1, 0, 0, 1, 0) (1, 1, 1, 1, 1, 1, 1, 1) (1, 0, 1, 0, 0, 1, 0, 1) (1, 0, 1, 1, 0, 1, 0, 0) (1, 0, 0, 1, 0, 1, 1, 0) For the initial form solutions we used the blackbox solver of PHCpack. Symbolic manipulations for the computation of the second term of the Puiseux series were done with Sage. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

23 an initial form system The pretropism v = (1, 1, 0, 1, 0, 0, 1, 0) defines x 1 + x 6 = 0 x 1 x 2 + x 5 x 6 + x 6 x 7 = 0 x 4 x 5 x 6 + x 5 x 6 x 7 = 0 x 0 x 1 x 6 x 7 + x 4 x 5 x 6 x 7 = 0 in v (f)(x) = x 0 x 1 x 2 x 6 x 7 + x 0 x 1 x 5 x 6 x 7 = 0 x 0 x 1 x 2 x 5 x 6 x 7 + x 0 x 1 x 4 x 5 x 6 x 7 + x 1 x 2 x 3 x 4 x 5 x 6 = 0 x 0 x 1 x 2 x 4 x 5 x 6 x 7 + x 1 x 2 x 3 x 4 x 5 x 6 x 7 = 0 x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 1 = 0 v defines the unimodular coordinate transformation: x 0 = y 0, x 1 = y 1 /y 0, x 2 = y 2, x 3 = y 0 y 3, x 4 = y 4, x 5 = y 5, x 6 = y 6 /y 0, x 7 = y 7. Using the new coordinates, we transform in v (f)(x). Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

24 the transformed initial form system in v (f)(y) = y 1 + y 6 = 0 y 1 y 2 + y 5 y 6 + y 6 y 7 = 0 y 4 y 5 y 6 + y 5 y 6 y 7 = 0 y 4 y 5 y 6 y 7 + y 1 y 6 y 7 = 0 y 1 y 2 y 6 y 7 + y 1 y 5 y 6 y 7 = 0 y 1 y 2 y 3 y 4 y 5 y 6 + y 1 y 2 y 5 y 6 y 7 + y 1 y 4 y 5 y 6 y 7 = 0 y 1 y 2 y 3 y 4 y 5 y 6 y 7 + y 1 y 2 y 4 y 5 y 6 y 7 = 0 y 1 y 2 y 3 y 4 y 5 y 6 y 7 1 = 0 Solving in v (f)(y), we obtain 8 solutions (all in the same orbit), e.g.: y 0 = t, y 1 = I, y 2 = 1 2 I 2, y 3 = 1, y 4 = 1+I, y 5 = I 2, y 6 = I, y 7 = 1 I, I = 1. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

25 developing a series for the solution Taking solution at infinity, we build a series of the form: y 0 = t y 1 = I + c 1 t y 2 = 1 2 I 2 + c 2t y 3 = 1+c 3 t y 4 = 1+I + c 4 t y 5 = I 2 + c 5t y 6 = I + c 6 t y 7 = ( 1 I)+c 7 t Plugging series form into transformed system, collecting all coefficients of t 1, solving yields c 1 = 1 I c 2 = 1 2 c 3 = 0 c 4 = 1 c 5 = 1 2 c 6 = 1+I c 7 = 1 The second term in the series, still in the transformed coordinates: y 0 = t y 1 = I +( 1 I)t y 2 = 1 2 I t y 3 = 1 y 4 = 1+I t y 5 = I t y 6 = I +(1+I)t y 7 = ( 1 I)+t Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

26 degree computations Definition (Branch Degree) Let v = (v 0, v 1,..., v n 1 ) be a tropism and let R be the set of initial roots of the initial form system in v (f)(y) = 0. n 1 Then the degree of the branch is #R max v i min n 1 v i i=0 i=0. Tropisms, their cyclic permutations, and degrees: (1, 1, 1, 1, 1, 1, 1, 1) 8 2 = 16 (1, 1, 0, 1, 0, 0, 1, 0) (1, 0, 0, 1, 0, 1, 1, 0) = 32 (1, 0, 1, 0, 0, 1, 0, 1) (1, 0, 1, 1, 0, 1, 0, 0) = 32 (1, 0, 1, 1, 0, 1, 0, 0) (1, 0, 1, 0, 0, 1, 0, 1) = 32 (1, 0, 0, 1, 0, 1, 1, 0) (1, 1, 0, 1, 0, 0, 1, 0) = 32 TOTAL = is the degree of the solution curve of the cyclic 8-root system. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

27 an initial form of cyclic 9-roots v 0 = (1, 1, 2, 1, 1, 2, 1, 1, 2) and v 1 = (0, 1, 1, 0, 1, 1, 0, 1, 1) define the initial form system x 2 + x 5 + x 8 = 0 x 0 x 8 + x 2 x 3 + x 5 x 6 = 0 x 0 x 1 x 2 + x 0 x 1 x 8 + x 0 x 7 x 8 + x 1 x 2 x 3 + x 2 x 3 x 4 + x 3 x 4 x 5 + x 4 x 5 x 6 + x 5 x 6 x 7 + x 6 x 7 x 8 = 0 x 0 x 1 x 2 x 8 + x 2 x 3 x 4 x 5 + x 5 x 6 x 7 x 8 = 0 x 0 x 1 x 2 x 3 x 8 + x 0 x 5 x 6 x 7 x 8 + x 2 x 3 x 4 x 5 x 6 = 0 x 0 x 1 x 2 x 3 x 4 x 5 + x 0 x 1 x 2 x 3 x 4 x 8 + x 0 x 1 x 2 x 3 x 7 x 8 + x 0 x 1 x 2 x 6 x 7 x 8 + x 0 x 1 x 5 x 6 x 7 x 8 + x 0 x 4 x 5 x 6 x 7 x 8 + x 1 x 2 x 3 x 4 x 5 x 6 + x 2 x 3 x 4 x 5 x 6 x 7 + x 3 x 4 x 5 x 6 x 7 x 8 = 0 x 0 x 1 x 2 x 3 x 4 x 5 x 8 + x 0 x 1 x 2 x 5 x 6 x 7 x 8 + x 2 x 3 x 4 x 5 x 6 x 7 x 8 = 0 x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 8 + x 0 x 1 x 2 x 3 x 5 x 6 x 7 x 8 + x 0 x 2 x 3 x 4 x 5 x 6 x 7 x 8 = 0 x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 1 = 0 Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

28 the unimodular transformation x = y M M = x 0 = y 0 x 1 = y 0 y 1 x 2 = y 2 0 y 1 1 y 2 x 3 = y 0 y 3 x 4 = y 0 y 1 y 4 x 5 = y 2 0 y 1 1 y 5 x 6 = y 0 y 6 x 7 = y 0 y 1 y 7 x 8 = y 2 0 y 1 1 y 8 We use the coordinate change to transform the initial form system and the original cyclic 9-roots system. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

29 the transformed initial form system y 2 + y 5 + y 8 = 0 y 2 y 3 + y 5 y 6 + y 8 = 0 y 2 y 3 y 4 + y 3 y 4 y 5 + y 4 y 5 y 6 + y 5 y 6 y 7 + y 6 y 7 y 8 + y 2 y 3 + y 7 y 8 + y 2 + y 8 = 0 y 2 y 3 y 4 y 5 + y 5 y 6 y 7 y 8 + y 2 y 8 = 0 y 2 y 3 y 4 y 5 y 6 + y 5 y 6 y 7 y 8 + y 2 y 3 y 8 = 0 y 2 y 3 y 4 y 5 y 6 y 7 + y 3 y 4 y 5 y 6 y 7 y 8 + y 2 y 3 y 4 y 5 y 6 + y 4 y 5 y 6 y 7 y 8 + y 2 y 3 y 4 y 5 + y 2 y 3 y 4 y 8 + y 2 y 3 y 7 y 8 + y 2 y 6 y 7 y 8 + y 5 y 6 y 7 y 8 = 0 y 3 y 4 y 6 y 7 + y 3 y 4 + y 6 y 7 = 0 y 4 y 7 + y 4 + y 7 = 0 y 2 y 3 y 4 y 5 y 6 y 7 y 8 1 = 0 A solution is y 2 = 1 2 3I 2, y 3 = I 2, y 4 = I 2, y 5 = 1, y 6 = 1 2 3I 2, y 7 = 1 2 3I 2, y 8 = I 2, where I = 1. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

30 an exact representation of a two dimensional set y 0 = t 1 x 0 = t 1 x 0 = y 0 x 1 = y 0 y 1 x 2 = y 2 0 y 1 1 y 2 x 3 = y 0 y 3 x 4 = y 0 y 1 y 4 x 5 = y 2 0 y 1 1 y 5 x 6 = y 0 y 6 x 7 = y 0 y 1 y 7 x 8 = y 2 0 y 1 1 y 8 y 1 = t 2 y 2 = 1 2 3I 2 y 3 = I 2 y 4 = I 2 y 5 = 1 y 6 = 1 2 3I 2 y 7 = 1 2 3I 2 y 8 = I 2 x 1 = t 1 t 2 x 2 = t 2 1 t 1 2 ( 1 2 3I 2 ) x 3 = t 1 ( I 2 ) x 4 = t 1 t 2 ( 1 3I ) x 5 = t 2 1 t 1 2 x 6 = t 1 ( 1 3I 2 2 ) x 7 = t 1 t 2 ( 1 2 3I 2 ) x 8 = t 2 1 t 1 2 ( I 2 ) Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

31 a tropical interpretation of Backelin s Lemma Denoting by u = e i2π/3 the primitive third root of unity, u 3 1 = 0: x 0 = t 0 x 1 = t 0 t 1 x 2 = t 2 0 t 1 1 u2 x 3 = t 0 u x 4 = t 0 t 1 u x 5 = t 2 0 t 1 1 x 6 = t 0 u 2 x 7 = t 0 t 1 u 2 x 8 = t 2 0 t 1 1 u. Introducing new variables y 0 = t 0, y 1 = t 0 t 1, and y 2 = t 2 0 t 1 1 u2 : x 0 = y 0 x 1 = y 1 x 2 = y 2 x 3 = y 0 u x 4 = y 1 u x 5 = y 2 u x 6 = y 0 u 2 x 7 = y 1 u 2 x 8 = y 2 u 2 which modulo y 3 0 y 3 1 y 3 2 u9 1 = 0 satisfies by plain substitution the cyclic 9-roots system, as in the proof of Backelin s Lemma, see J.C. Faugère. Finding all the solutions of Cyclic 9 using Gröbner basis techniques. In Computer Mathematics - Proceedings of the Fifth Asian Symposium (ASCM 2001), pages World Scientific, Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

32 degree computations For u 3 = 1, our representation of the solution set is x 0 = t 0 x 1 = t 0 t 1 x 2 = t 2 0 t 1 1 u2 x 3 = t 0 u x 4 = t 0 t 1 u x 5 = t 2 0 t 1 1 x 6 = t 0 u 2 x 7 = t 0 t 1 u 2 x 8 = t 2 0 t 1 1 u. We compute the degree of the surface using two random hyperplanes: { α1 t 0 +α 2 t 0 t 1 +α 3 t 2 0 t 1 1 = 0 α 4 t 0 +α 5 t 0 t 1 +α 6 t 2 0 t 1 1 = 0, α 1,α 2,...,α 6 C. Simplifying, the system becomes { t 2 0 t 1 1 β 1 = 0 t 1 β 2 = 0, β 1,β 2 C. There are 3 solutions, so we have a cubic surface of cyclic 9 roots. Applying symmetry, we find an orbit of 6 cubic surfaces. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

33 cyclic m 2 -roots Proposition For n = m 2, there is an (m 1)-dimensional set of cyclic n-roots, represented exactly as x km+0 = u k t 0 x km+1 = u k t 0 t 1 x km+2 = u k t 0 t 1 t 2 x km+m 2 x km+m 1. = u k t 0 t 1 t 2 t m 2 = u k t m+1 0 t m+2 1 t 2 m 3 t 1 m 2 for k = 0, 1, 2,...,m 1 and u k = e i2kπ/m. The degree of this solution set equals m. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34

34 conclusion Promising results on the cyclic n-roots problem give a proof of concept for a new polyhedral method to compute algebraic sets. Papers available via jan: J. Verschelde. Polyhedral methods in numerical algebraic geometry. In Interactions of Classical and Numerical Algebraic Geometry, volume 496 of Contemporary Mathematics, pages AMS, D. Adrovic and J. Verschelde. Polyhedral methods for space curves exploiting symmetry. arxiv: v1. D. Adrovic and J. Verschelde. Computing Puiseux series for algebraic surfaces. Proceedings of ISSAC 2012, to appear. Version of PHCpack solves binomial systems with phc -b. Jan Verschelde (UIC) Newton polytopes, tropisms, Puiseux series SIAM DM / 34