Condition Length and Complexity for the Solution of Polynomial Systems Diego Armentano, Carlos Beltrán, Peter Bürgisser, Felipe Cucker & Michael Shub Foundations of Computational Mathematics The Journal of the Society for the Foundations of Computational Mathematics ISSN 1615-3375 Found Comput Math DOI 10.1007/s10208-016-9309-9 1 23 Your article is protected by copyright and all rights are held exclusively by SFoCM. This e- offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Found Comput Math DOI 10.1007/s10208-016-9309-9 Condition Length and Complexity for the Solution of Polynomial Systems Diego Armentano1 · Carlos Beltrán2 · Peter Bürgisser3 · Felipe Cucker4 · Michael Shub5 Received: 20 July 2015 / Revised: 11 January 2016 / Accepted: 2 February 2016 © SFoCM 2016 Abstract Smale’s 17th problem asks for an algorithm which finds an approximate zero of polynomial systems in average polynomial time (see Smale in Mathematical problems for the next century, American Mathematical Society, Providence, 2000). The main progress on Smale’s problem is Beltrán and Pardo (Found Comput Math Communicated by Teresa Krick. Diego Armentano partially supported by Agencia Nacional de Investigación e Innovación (ANII), Uruguay, and by CSIC group 618. Carlos Beltrán partially suported by the research projects MTM2010-16051 and MTM2014-57590-P from Spanish Ministry of Science MICINN. Peter Bürgisser partially funded by DFG Research Grant BU 1371/2-2. Felipe Cucker partially funded by a GRF Grant from the Research Grants Council of the Hong Kong SAR (Project Number CityU 100813). B Michael Shub [email protected] Diego Armentano [email protected] Carlos Beltrán [email protected] Peter Bürgisser [email protected] Felipe Cucker [email protected] 1 Universidad de La República, Montevideo, Uruguay 2 Universidad de Cantabria, Santander, Spain 3 Technische Universität Berlin, Berlin, Germany 4 City University of Hong Kong, Kowloon Tong, Hong Kong 5 City University of New York, New York, NY, USA 123 Author's personal copy Found Comput Math 11(1):95–129, 2011) and Bürgisser and Cucker (Ann Math 174(3):1785–1836, 2011). In this paper, we will improve on both approaches and prove an interesting intermediate result on the average value of the condition number. Our main results are Theorem 1 on the complexity of a randomized algorithm which improves the result of Beltrán and Pardo (2011), Theorem 2 on the average of the condition number of polynomial systems which improves the estimate found in Bürgisser and Cucker (2011), and Theorem 3 on the complexity of finding a single zero of polynomial systems. This last theorem is similar to the main result of Bürgisser and Cucker (2011) but relies only on homotopy methods, thus removing the need for the elimination theory methods used in Bürgisser and Cucker (2011). We build on methods developed in Armentano et al. (2014). Keywords Polynomial systems · Homotopy methods · Complexity estimates Mathematics Subject Classification Primary 65H10 · 65H20 · Secondary 58C35 1 Introduction Homotopy or continuation methods to solve a problem which might depend on parame- ters start with a problem instance and known solution and try to continue the solution along a path in parameter space ending at the problem we wish to solve. We recall how this works for the solutions of polynomial systems using a variant of Newton’s method to accomplish the continuation. Let Hd be the complex vector space of degree d complex homogeneous polynomials + α = (α ,...,α ) ∈ Nn+1 n α = in n 1 variables. For 0 n satisfying j=0 j d, and the α α α = 0 ··· n H α, β := monomial z z0 zn , the Weyl Hermitian structure on d makes z z 0, for α = β and − − d 1 d! 1 zα, zα:= = . α α0!···αn! ( ) = ( ,..., ) H = n H Now for d d1 dn ,welet (d) k=1 dk . This is a complex vector space of dimension n n + d N := i . n i=1 That is, N is the size of a system f ∈ H(d), understood as the number of coefficients needed to describe f . We endow H(d) with the product Hermitian structure n f, g:= fi , gi , k=1 123 Author's personal copy Found Comput Math where f = ( f1,..., fn), and g = (g1,...,gn). This Hermitian structure is sometimes called the Weyl, Bombieri-Weyl, or Kostlan Hermitian structure. It is invariant under unitary substitution f → f ◦ U −1, where U is a unitary transformation of Cn+1 (see [9, p. 118] for example). On Cn+1, we consider the usual Hermitian structure n x, y:= xk yk. k=0 Given 0 = ζ ∈ Cn+1,letζ ⊥ denote the Hermitian complement of ζ , ⊥ + ζ := {v ∈ Cn 1 :v,ζ=0}. For any nonzero ζ ∈ Cn+1, the subspace ζ ⊥ is a model for the tangent space, + + Tζ P(Cn 1), of the projective space P(Cn 1) at the equivalence class of ζ (which + we also denote by ζ ). The space Tζ P(Cn 1) inherits an Hermitian structure from ·, · given by v,w v,w := . ζ ζ,ζ See, for example, [9, Sec. 12.2] for more details on this standard metric structure of P(Cn+1). The group of unitary transformations U acts naturally on Cn+1 by ζ → Uζ for U ∈ U, and the Hermitian structure of Cn+1 is invariant under this action. n+1 A zero of the system of equations f is a point x ∈ C such that fi (x) = 0, i = 1,...,n. If we think of f as a mapping f : Cn+1 → Cn, it is a point x such that f (x) = 0. ∈ H For a generic system (that is, for a Zariski open set of f (d)), Bézout’s theorem D := n states that the set of zeros consists of k=1 dk complex lines through 0. These D lines are D points in projective space P(Cn+1). So our goal is to approximate one of these points, and we will use homotopy or continuation methods. These methods for the solution of a system f ∈ H(d) proceed as follows. Choose n+1 g ∈ H(d) and a zero ζ ∈ P(C ) of g (we denote by the same symbol an affine point and its projective class). Connect g to f by a path ft ,0≤ t ≤ 1, in H(d) such that f0 = g, f1 = f , and try to continue ζ0 = ζ to ζt such that ft (ζt ) = 0, so that f1(ζ1) = 0(see[7] for details or [12] for a complete discussion). So homotopy methods numerically approximate the path ( ft ,ζt ).Onewayto accomplish the approximation is via (projective) Newton’s method. Given an approx- imation xt to ζt , define := ( ), xt+t N ft+t xt n+1 where for h ∈ H(d) and y ∈ P(C ) we define the projective Newton’s method Nh(y) following [17]: ( ) := − ( ( )| )−1 ( ). Nh y y Dh y y⊥ h y 123 Author's personal copy Found Comput Math P(Cn+1) ( )| Note that Nh is defined on at those points where Dh y y⊥ is invertible. That xt is an approximate zero of ft with associated (exact) zero ζt means that the sequence of Newton iterations N k (x ) converges immediately and quadratically to ζ . ft t t Let us assume that { ft }t∈[0,1] is a path in the sphere S(H(d)) := {h ∈ H(d) : 1 h =1}. The main result of [16] is that the tk may be chosen so that t0 = 0, = + = ,..., = tk tk−1 tk for k 1 K with tK 1, such that for all k, xtk is an approximate ζ zero of ftk with associated zero tk , and the number K of steps can be bounded as follows: 1 3/2 ˙ ˙ K = K ( f, g,ζ)≤ CD μ( ft ,ζt ) ( ft , ζt ) dt. (1.1) 0 Here C is a universal constant, D = maxi di , √ −1 d −1 f (Df(ζ )|ζ ⊥ ) diag( ζ i di ) if Df(ζ )|ζ ⊥ is invertible μ( f,ζ):= ∞ otherwise n+1 is the condition number of f ∈ H(d) at ζ ∈ P(C ),diag(v) is the diagonal matrix whose diagonal entries are the coordinates of the vector v), and ( f˙ , ζ˙ ) =( f˙ 2 + ζ˙ 2 )1/2 t t t t ζt is the norm of the tangent vector to the curve in ( ft ,ζt ). The result in [16]isnotfully constructive, but specific constructions have been given, see [3] and [14], and even programmed [4]. These constructions are similar to those given in [20] and [2] (this last, for the eigenvalue-eigenvector problem case). The constructive versions cited above have slightly different criteria to choose the step length, which is the backbone of the continuation algorithm. However, all these algorithms satisfy a unitary invariance in the sense that if U is a unitary matrix of size n + 1 then ∗ ∗ K ( f, g,ζ)= K ( f ◦ U , g ◦ U , Uζ). (1.2) The right-hand side in expression (1.1) is known as the condition length of the path ( ft ,ζt ). We will call (1.1)thecondition length estimate of the number of steps.