Computing all Space Curve Solutions of Polynomial Systems by Polyhedral Methods
CComputing all Space Curve Solutions ofPolynomial Systems by Polyhedral Methods ∗ Nathan Bliss Jan VerscheldeUniversity of Illinois at ChicagoDepartment of Mathematics, Statistics, and Computer Science851 S. Morgan Street (m/c 249), Chicago, IL 60607-7045, USA { nbliss2,janv } @uic.edu Abstract
A polyhedral method to solve a system of polynomial equations ex-ploits its sparse structure via the Newton polytopes of the polynomials.We propose a hybrid symbolic-numeric method to compute a Puiseux se-ries expansion for every space curve that is a solution of a polynomialsystem. The focus of this paper concerns the difficult case when the lead-ing powers of the Puiseux series of the space curve are contained in therelative interior of a higher dimensional cone of the tropical prevariety. Weshow that this difficult case does not occur for polynomials with genericcoefficients. To resolve this case, we propose to apply polyhedral endgames to recover tropisms hidden in the tropical prevariety.
Key words and phrases.
Newton polytope, polyhedral end game, poly-hedral method, polynomial system, Puiseux series, space curve, tropicalbasis, tropical prevariety, tropism.
In this paper we consider the application of polyhedral methods to computeseries for all space curves defined by a polynomial system. Polyhedral methodscompute with the Newton polytopes of the system. The
Newton polytope of apolynomial is defined as the convex hull of the exponents of the monomials thatappear with a nonzero coefficient.If we start the development of the series where the space curve meets thefirst coordinate plane, then we compute Puiseux series. Collecting for eachcoordinate the leading exponents of a Puiseux series gives what is called a tropism . If we view a tropism as a normal vector to a hyperplane, then wesee that there are hyperplanes with this normal vector that touch every Newton ∗ This material is based upon work supported by the National Science Foundation underGrant No. 1440534. a r X i v : . [ c s . S C ] J un olytope of the system at an edge or at a higher dimensional face. A vectornormal to such a hyperplane is called a pretropism . While every tropism is apretropism, not every pretropism is a tropism.In this paper we investigate the application of a polyhedral method to com-pute all space curve solutions of a polynomial system. The method starts fromthe collection of all pretropisms, which are regarded as candidate tropisms. Forthe method to work, we focus on the following questions. Problem Statement.
Given that only the space curves are of interest, canwe ignore the higher dimensional cones of pretropisms? In particular, if sometropisms lie in the interior of higher dimensional cones of pretropisms, is it thenstill possible to compute Puiseux series solutions for all space curves?
Related Work.
In symbolic computation, new elimination algorithms forsparse systems with positive dimensional solution sets are described in [8]. Trop-ical resultants are computed in [13]. Related polyhedral methods for sparsesystems can be found in [10, 15]. Conditions on how far a Puiseux series shouldbe expanded to decide whether a point is isolated are given in [6]. The authorsof [12] propose numerical methods for tropical curves. Polyhedral methods tocompute tropical varieties are outlined in [4] and implemented in Gfan [14]. Thebackground on tropical algebraic geometry is in [16].Algorithms to compute the tropical pre variety are presented in [21]. For pre processing purposes, the software of [21] is useful. However, the focus on thispaper concerns the tropical variety for which Gfan [14] provides a tropical basis.Therefore, our computational experiments with computer algebra methods areperformed with Gfan and not with the software of [21].
Organization and Contributions.
In the next section we illustrate the ad-vantages of looking for Puiseux series as solutions of polynomial systems. Thenwe motivate our problem with some illustrative examples. Relating the tropicalprevariety to a recursive formula to compute the mixed volume characterizesthe generic case, in which the tropical prevariety suffices to compute all spacecurve solutions. With polyhedral end games we can recover the tropisms con-tained in higher dimensional cones of the tropical prevariety. Finally we givesome experimental results and timings.
When we work with Puiseux series we apply a hybrid method, combining exactand approximate calculations. Figure 1 shows the plot, in black, of Viviani’scurve, defined as the intersection of the sphere f = x + x + x − g = ( x − + x − v = (2 , , f and g respectively as x − x − x . For traditional Puiseux series, one wouldchoose to set x = 1, obtaining the four solutions (1 , ±√ , ±
2) and leadingterms ( t , ±√ t, ± x = 2, we obtain rational coefficients2 igure 1: Viviani’s curve with improving Puiseux series approximations, la-belled with the number of terms used to plot each one.and the following partial expansion: x x x = t t − t − t − t − t − t − t − t − t . (1)The plot of several Puiseux approximations to Viviani’s curve is shown in grayin Figure 1.If we shift the Viviani example so that its self-intersection is at the origin,we obtain the following: f ( x ) = (cid:26) x + x + x + 4 x = 0 x + x + 2 x = 0 (2)An examination of the first few terms of the Puiseux series expansion for thissystem, combined with the On-Line Encyclopedia of Integer Sequences [17] andsome straightforward algebraic manipulation, allows us to hypothesize the fol-lowing exact parameterization of the variety: x x x = − t t √ − t − t − t . (3)We can confirm that this is indeed right via substitution. While this methodis of course not possible in general, it does provide an example of the potentialusefulness of Puiseux series computations for some examples.3 Assumptions and Setup
Our object of study is space curves, by which we mean 1-dimensional varietiesin C n . Because Puiseux series computations take one variable to be a freevariable, we require that the curves not lie inside V ( (cid:104) x i (cid:105) ) for some i ; withoutloss of generality we choose to use the first variable. Some results require thatthe curve be in Noether position with respect to x , meaning that the degreeof the variety is preserved under intersection with x = λ for a generic λ ∈ C .It is of course possible to apply a random coordinate transformation to obtainNoether position, but we then lose the sparsity of the system’s exponent supportstructure, which is what makes polyhedral methods effective. In this section we illustrate the problem our paper addresses with some sim-ple examples, first in 3-space, and then with a family of space curves in anydimensional space.
Our first running example is the system f ( x ) = (cid:26) x x − x x − x + x = 0 x − x x − x x − x − x = 0 (4)which has an irreducible quartic and the second coordinate axis (0 , x ,
0) as itssolutions. Because the line lies in the first coordinate plane x = 0, the systemis not in Noether position with respect to the first variable. Therefore, ourmethods will ignore this part of the solution set. The algorithms of [7] can beapplied to compute components inside coordinate planes. Computing a primarydecomposition yields the following alternative, which lacks the portion in thefirst coordinate plane: ˜f ( x ) = x x − x x − x + x x x − x − x x + x + x − x − x x − x − x x − x − x (5)The tropical prevariety contains the rays (2 , , , , , , x = 0, rays that have azero or negative value for their first coordinate have been discarded. The trop-ical variety however contains the ray (3 , ,
1) instead of (2 , , x x x = t t − t − t + 27 t + 36 t − t − t − t + 18 t + 162 t . (6)4his ray is a positive combination of (2 , ,
1) and (1 , , This problem can also occur in arbitrary dimensions, as seen in the class ofexamples f ( x ) = x − x + x + x + · · · + x n = 0 x + x + x + x + · · · + x n = 0 x + x + x + x + · · · + x n = 0... x n − + x + x + x + · · · + x n = 0 . (7)The ray (1 , , , . . . ,
1) is a 1-dimensional cone of its prevariety since it is normalto a facet of each polytope, namely the linear portion of each polynomial. It isnot, however, in the tropical variety, since the initial form system (as it will bedefined in Section 5) contains the monomial x . This hiding of tropisms in the higher dimensional cones of the prevariety isproblematic, as finding the tropical variety may require more expensive symboliccomputations. For a comparison between various approaches see Section 9.Fortunately, this problem does not occur in general, as the next result willshow. But first, a few definitions.
Definition 5.1.
We write a polynomial f with support set A as f ( x ) = (cid:88) a ∈ A c a x a , c a ∈ C ∗ , x a = x a x a · · · x a n n . (8)The initial form of f with respect to v is thenin v f ( x ) = (cid:88) a ∈ in v A c a x a , (9)where in v A = { a ∈ A | (cid:104) a , v (cid:105) = min b ∈ A (cid:104) b , v (cid:105) } .The initial form of a tuple of polynomials is the tuple of the initial forms ofthe polynomials in the tuple. Definition 5.2.
For f ∈ C [ x ], I an ideal in C [ x ] and v ∈ R n , we define the initial ideal in v ( I ) as the ideal generated by { in v ( f ) : f ∈ I } . Definition 5.3.
For I = (cid:104) f , . . . , f m (cid:105) ⊂ C [ x ] an ideal, the tropical prevariety isthe set of v ∈ R n for which in v ( f i ) is not a monomial for any i . The tropicalvariety is the set of v ∈ R n for which in v ( f ) is not a monomial for any f ∈ I .5 roposition 5.4. For n equations in n + 1 unknowns with generic coefficients,the set of ray generators of the tropical prevariety contains the tropical variety. It is important to note that our notion of generic here refers to the coeffi-cients, and not to generic tropical varieties as seen in [18] which are tropicalvarieties of ideals under a generic linear transformation of coordinates.The tropical prevariety always contains the tropical variety. We simply wantto show that all of the rays of the tropical variety show up in the prevarietyas ray generators, and not as members of the higher-dimensional cones. Let I = (cid:104) p , . . . , p n (cid:105) ⊆ C [ x , . . . , x n ], and let w be a ray in the tropical prevarietybut not one of its ray generators. We want to show that w is not in the tropicalvariety, or equivalently that in w ( I ) contains a monomial. We will do so byshowing that I w := (cid:104) in w ( p ) , . . . , in w ( p n ) (cid:105) contains a monomial, which sufficessince this ideal is contained in in w ( I ).Suppose I w contains no monomial. Then ( x x · · · x n ) k / ∈ I w for any k . ByHilbert’s Nullstellensatz V := V ( I w ) (cid:42) V ( x x · · · x n ), i.e. V is not contained inthe union of the coordinate hyperplanes. Then there exists a = ( a , . . . , a n ) ∈ V such that all coordinates of a are all nonzero. Since w lies in the interior a coneof dimension at least 2, the generators of I w are homogeneous with respect to atleast two linearly independent rays u and v . Thus ( λ u µ v a , . . . , λ u n µ v n a n ) ∈ V for all λ, µ ∈ C \ { } where the u i , v i are the components of u and v , and V contains a toric surface. If we intersect with a random hyperplane, by Bern-stein’s theorem B [3] the result is a finite set of points, with the possibility ofadditional components that must be contained in the coordinate planes. Hence V can contain no surface outside of the coordinate planes, and we have a con-tradiction. We will show that the tropical prevariety provides an upper bound for the degreeof the solution curve. The inner product of a point a with a vector v is denotedas (cid:104) a , v (cid:105) = a v + a v + · · · + a n v n . Lemma 6.1.
Consider an ( n − -tuple of Newton polytopes P = ( P , P , . . . , P n − ) in n -space. Let E be the edge spanned by (1 , , . . . , and (0 , , . . . , . Themixed volume of ( P , E ) equals V n ( P , E ) = (cid:88) v v V n − (in v P ) , (10) where v ranges over all rays in the tropical prevariety of P with v > , nor-malized so that gcd( v ) = gcd( v , v , . . . , v n ) = 1 , and in v P = (in v P , in v P , . . . , in v P n − ) , where in v P k is the face with support vector v , formally expressed as in v P k = { a ∈ P k | (cid:104) a , v (cid:105) = max a ∈ P k (cid:104) a , v (cid:105) } . (11)6e apply the following recursive formula [20] for the mixed volume V n ( P , E ) = (cid:88) v ∈ Z n gcd( v ) = 1 p E ( v ) V n − (in v P ) , (12)where p E is the support function of the edge E : p E ( v ) = max e ∈ E (cid:104) e , v (cid:105) (13)and in v P = (in v P , in v P , . . . , in v P n − ), wherein v P k = { a ∈ P k | (cid:104) a , v (cid:105) = p k ( v ) } , (14)with p k the support function of the polytope P k .Because the edge E contains (0 , , . . . , p E ( v ) ≥ p E ( v ) = 0 when v ≤
0. Only those rays for which v > V n ( P , E ). We have then p E ( v ) = v .The mixed volume of a tuple of polytopes equals zero if one of the polytopesconsists of only one vertex. The rays in the tropical prevariety contain all vectorsfor which in v ( P k ) is an edge or a higher dimensional face. These are the rays v for which V n − (in v P ) > C ∗ = C \ { } . Lemma 6.2.
Consider the system f ( x ) = , f = ( f , f , . . . , f n − ) with P = ( P , P , . . . , P n − ) where P k is the Newton polytopes of f k . If the system isin Noether position with respect to x , then the degree of the space curve definedby f ( x ) = is bounded by V n ( P , E ) . This result is a version of Lemma 2.3 from [15].The proof of the lemma follows from the application of Bernshtein’s theo-rem [3] to the system (cid:26) f ( x ) = x = γ, γ ∈ C ∗ . (15)By the assumption of Noether position, there will be as many solutions to thissystem as the degree of the space curve defined by f ( x ) = . The theoremof Bernshtein states that the mixed volume bounds the number of solutions in( C ∗ ) n .Formula (12) appears in the constructive proof of Bernshtein’s theorem [3]and was implemented in the polyhedral homotopies of [25]. For systems withcoefficients that are sufficiently generic, the mixed volumes provide an exactroot count. 7 heorem 6.3. Let f ( x ) = be a polynomial system of n − equations in n unknowns, with sufficiently generic coefficients. Assume the space curve definedby f ( x ) = is in Noether position with respect to the first variable. Then all rays v with v > in the tropical prevariety of f lead to Puiseux series expansionsfor the space curve defined by f ( x ) = . Moreover, the degree of the space curveis the sum of the degrees of the Puiseux series. We illustrate the application of polyhedral methods to the motivating ex-amples.
Example 6.4.
As a verification on the first motivating example (4), we considerthe rays (2 , , , , , ,
1) of its tropical prevariety. The initial formof f in (4) w.r.t. to the ray (2 , ,
1) isin (2 , , f ( x ) = (cid:26) − x x − x + x = 0 − x x − x − x = 0 . (16)To count the number of solutions of in (2 , , f ( x ) = we apply a unimodularcoordinate transformation, x = y U : U = x = y y x = y y x = y (17)which leads to the systemin (2 , , f ( y ) = (cid:26) − y y − y + y y = 0 − y y − y − y y = 0 . (18)After removing the common factor y , we see that this system has one solutionfor generic choices of the coefficients. As 2 is the first coordinate of (2 , , , ,
0) and (1 , ,
1) each contribute one to the degree, and sowe recover the degree four of the solution curve.
Example 6.5.
For the family of systems in (7), consider the curve in 4-space: f ( x ) = x − x + x + x + x = 0 x + x + x + x + x = 0 x + x + x + x + x = 0 . (19)For the tropism v = (2 , , , v f ( x ) = x + x + x = 0 x + x + x = 0 x + x + x = 0 . (20)This tropism is in the interior of the cone in the tropical prevariety spannedby v = (1 , , ,
1) and v = (1 , , , v ( I ) has a mixed volume of one and in v ( I ) has amixed volume of three, so for generic coefficients we again recover the degree ofthe solution curve. 8 Current Approaches
In [4] a method is given for computing the tropical variety of an ideal I defining acurve. It involves appending witness polynomials from I to a list of its generatorssuch that for this new set, the tropical prevariety equals the tropical variety.Such a set is called a tropical basis . Each additional polynomial rules out one ofthe cones in the original prevariety that does not belong in the tropical variety.As stated in [4] only finitely many additional polynomials are necessary, sincethe prevariety has only finitely many cones.The algorithm runs as follows. For each cone C in the tropical prevariety, wechoose a generic element w ∈ C . We check whether in w ( I ) contains a monomialby saturating with respect to m , the product of ring variables; the initial idealcontains a monomial if and only if this saturation ideal is equal to (1). If in w ( I )does not contain a monomial, the cone C belongs in our tropical variety. If itdoes, we check whether m i ∈ I for increasing values of i until we find a monomial m (cid:48) ∈ in w ( I ). Finally, we append m (cid:48) − h to our list of basis elements, where h is the reduction of m with respect to a Gr¨obner basis of I under any monomialorder that refines w . For w to define a global monomial order, and thus allowa Gr¨obner basis, it may be necessary to homogenize the ideal first.Bounding the complexity of this algorithm is beyond the scope of this paper,but for each cone it requires computing a Gr¨obner basis of I as well as another(possibly faster) basis when calculating the saturation to check if the initialideal contains a monomial. In some cases we may only be concerned abouttropisms hiding in a particular higher-dimensional cone of the prevariety, suchas with our running example (7). Here it is reasonable to perform only one stepof this algorithm, namely looking for a witness for a single cone, which couldbe significantly faster. However, this has the disadvantage of introducing more1-dimensional cones into the prevariety. More details, including some timingcomparisons, will be given in Section 9. A polyhedral end game [11] applies extrapolation methods to numerically esti-mate the winding number of solution paths defined by a homotopy. The leadingexponents of the Puiseux series are recovered via differences of the logarithmsof the magnitudes of the coordinates of the solution paths. Even in the case –as in our illustrative example – where the given polynomials contain insufficientinformation to compute all tropisms only from the prevariety, a polyhedral endgame manages to compute all tropisms. The setup is similar to that of [23],arising in a numerical study of the asymptotics of a space curve, defined by thesystem f ( x ) = : (cid:26) f ( x ) = tx + (1 − t )( x − γ ) = 0 , γ ∈ C \ { } , (21)9s t moves from 0 to 1, the hyperplane x = γ moves the coordinate planeperpendicular to the first coordinate axis.As t moves from 0 to 1, it is important to note that t will actually neverbe equal to one. In the polyhedral end games of [11], to estimate the windingnumber via extrapolation methods, the step size decreases in a geometric ratio.In particular, denoting the winding number by ω , for t = 1 − s ω , and 0 < r < s k = s r k , k = 0 , , . . . , starting at some s ≈ γ in (21) is a randomly generated complex number. This im-plies that for x = γ , the polynomial system in (21) for t = 0 has as manyisolated solutions (generic points on the space, eventually counted with multi-plicities) as the degree of the projection of the space curve onto the first coordi-nate plane. As long as t <
1, the points remain generic, although the numericalcondition numbers are expected to blow up as t approaches one.The deteriorating numerical ill conditioning can be mitigated by the useof multiprecision arithmetic. For example, condition numbers larger than 10 make results unreliable in double precision. In double double precision, muchhigher condition numbers can be tolerated, typically up to 10 , and this goes upto 10 for quad double precision. As we interpret the inverse of the conditionnumber as the distance to a singular solution, with multiprecision arithmeticwe can compute more points more accurately as needed in the extrapolation toestimate winding numbers.An additional difficulty arises when a path diverges to infinity, which mani-fests itself by a tropism with negative coordinates. A reformation of the problemin a weighted projective space corresponds to a unimodular coordinate trans-formation which uses the computed direction of the solution path. Towards theend of the path, this direction coincides with the tropism.The a posteriori verification of a polyhedral end game is similar to computinga Puiseux expansion starting at a pretropism. In this section we focus on the family of systems (7) with a tropism hidden ina higher dimensional cone of pretropisms. Classical families such as the cyclic n -roots problems appear not to have such hidden pretropisms, at least not forthe cases computed in [1, 2] and [19]. To substantiate the claim that finding the tropical variety is computationallyexpensive, we calculated tropical bases of the system (7) for various values of n .The symbolic computations of tropical bases was done with Gfan [14]. Timesare displayed in Figure 1. The computations were executed on an Intel XeonE5-2670 processor running RedHat Linux. As is clear from the table, as thedimension grows for this relatively simple system, computation time becomesprohibitively large. 10 able 1: Execution times, in seconds, of the computation of a tropical basisfor the system (7); averages of 3 trials.n 3 4 5 6 7time 0.052 0.306 2.320 33.918 970.331As mentioned in Section 7, an alternative to computing the tropical basisis to only calculate the witness polynomial for a particular cone of the tropicalprevariety. We implemented this algorithm in Macaulay2 [5] and applied it to (7)to cut down the cone generated by the rays (1 , , . . . ,
1) and (1 , , , . . . , , , . . . , Table 2:
Execution times in seconds of the computation of a witness poly-nomial for the cone generated by (1 , , . . . , , (1 , , . . . ,
0) of the system (7);averages of 3 trials. The third column lists the number of rays in the fan ob-tained by intersecting the original prevariety with the normal fan of the witnesspolynomial; since this can vary with the choice of random ray, we list valuesfrom several tries. dim time
The polyhedral end game was done with version 2.4.10 of PHCpack [22], up-graded with double double and quad double arithmetic, using QDlib [9]. Poly-hedral end games are also available via the Python interface of PHCpack, since11ersion 0.4.0 of phcpy [24].For the first motivating example (4) in 3-space, there are four solutionswhen x = γ . The tropism (3 , , Table 3:
Execution times on tracking d paths in n -space with a polyhedral endgame. The reported time is the elapsed CPU user time, in seconds. The lastcolumn represents the average time spent on one path. n d time time/d4 4 0.012 0.0035 8 0.035 0.0066 16 0.090 0.0077 32 0.243 0.0108 64 0.647 0.0139 128 1.683 0.01610 256 4.301 0.01711 512 7.507 0.01512 1024 27.413 0.027All directions computed with double precision at an accuracy of 10 − . Dou-ble precision sufficed to accurately compute the tropism (2 , , . . . ,
10 Conclusions
The tropical prevariety provides candidate tropisms for Puiseux series expan-sions of space curves. As shown in [1, 2] on the cyclic n -root problems, thepretropisms may directly lead to series developments for the positive dimensionalsolution sets. In this paper we studied cases where tropisms are in the relativeinterior of higher-dimensional cones of the tropical prevariety. If the tropicalprevariety contains a higher dimensional cone and Puiseux series expansion failsat one of the cone’s generating rays, then a polyhedral end game can recoverthe tropisms in the interior of that higher dimensional cone of pretropisms. Asour example shows, this takes drastically less time than computing the tropicalvariety via a tropical basis, especially as dimension grows. It is also faster thanfinding a witness polynomial for just that particular cone, and avoids the issueof adding rays to the tropical prevariety.12 eferences [1] D. Adrovic and J. Verschelde. Computing Puiseux series for algebraic sur-faces. In J. van der Hoeven and M. van Hoeij, editors, Proceedings ofthe 37th International Symposium on Symbolic and Algebraic Computation(ISSAC 2012) , pages 20–27. ACM, 2012.[2] D. Adrovic and J. Verschelde. Polyhedral methods for space curves ex-ploiting symmetry applied to the cyclic n -roots problem. In V.P. Gerdt,W. Koepf, E.W. Mayr, and E.V. Vorozhtsov, editors, Computer Algebra inScientific Computing, 15th International Workshop, CASC 2013, Berlin,Germany , volume 8136 of
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