Numerical Computations in Algebraic Geometry Jan Verschelde University of Illinois at Chicago Department of Mathematics, Statistics, and Computer Science http://www.math.uic.edu/ jan jan@math.uic.edu AMS MRC Computational Algebra and Convexity Snowbird, Utah, 21-27 June 28 Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 1 / 3
Outline 1 Numerically Solving Polynomial Systems 2 Zero Dimensional Solving homotopy continuation methods deflating isolated singularities 3 Numerical Irreducible Decomposition witness sets and cascades of homotopies monodromy grouping certified by linear traces 4 Numerical Primary Decomposition Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 2 / 3
Numerically Solving Polynomial Systems problems and methods in numerical algebraic geometry Numerical algebraic geometry applies numerical analysis to problems in algebraic geometry. Algebraic geometry studies solutions of polynomial systems, therefore the main problem is to solve polynomial systems. Numerical analysis is concerned with the efficiency and accuracy of mathematical algorithms using floating-point arithmetic. Three levels of solving a polynomial system: 1 focus on the isolated complex solutions 2 output contains also positive dimensional solution sets 3 a numerical primary decomposition Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 3 / 3
Why use numerical analysis? costs and benefits of the numerical approach Computational algebraic geometry mainly uses symbolic computing: resultants and Gröbner bases are well developed, already excellent software tools in computer algebra, exact computations are more trustworthy. Two benefits of numerical analysis: accepts approximate input, offers sensitivity analysis many real-world applications have approximate input data approximating algebraic numbers leads to speedups as well pleasingly parallel algorithms for high performance computing with Python possible to do interactive parallel computing Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 4 / 3
Three References most relevant for this talk 1 Tien-Yien Li: Numerical solution of polynomial systems by homotopy continuation methods. In Volume XI of Handbook of Numerical Analysis, pp. 29 34, 23. 2 Andrew J. Sommese and Charles W. Wampler: The Numerical Solution of Systems of Polynomials Arising in Engineering and Science. World Scientific, 25. 3 Anton Leykin: Numerical Primary Decomposition. arxiv:81.315v2 [math.ag] 29 May 28. To appear in the proceedings of ISSAC 28. Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 5 / 3
Zero Dimensional Solving specification of input and output Input: f (x) =, x =(x 1, x 2,...,x n ), f =(f 1, f 2,...,f n ) (C[x]) n. The polynomial system f (x) = has as many equations as unknowns. In the output of a numerical zero dimensional solver we expect to find two types of solutions in C n : 1 regular: Jacobian matrix is of full rank, 2 multiple: some isolated solutions coincide. But we may find also two other types of solutions: 1 at infinity: if we have fewer solutions than expected, 2 on component: if we have more solutions than expected. Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 6 / 3
Solving Simpler Systems to get to the expected number of isolated complex solutions How many isolated complex solutions do we expect? Apply the method of degeneration: 1 Bézout: deform equations into products of linear equations 2 Bernshteǐn, Kushnirenko, Khovanskiǐ: deform system into initial binomial systems Embed the target problem into a family of similar problems. Solve the simpler systems and follow paths of solutions starting at the solutions of the simpler systems to the solutions of the target problem. Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 7 / 3
Tracking Solution Paths using predictor-correct methods Given is a family of systems: h(x, t) =, a homotopy. A typical form of a homotopy to solve f (x) = is h(x, t) =(1 t)g(x)+tf(x) =, where g(x) = is a simpler good system. The parameter t is an artificial continuation parameter. Three key algorithmic ingredients: 1 Newton s method 2 predictor-corrector methods 3 endgames to deal with singularities Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 8 / 3
Puiseux Series in Endgames Solving f (x) = using start system g(x) =: h(x, t) =(1 t)g(x)+tf(x) =, t 1. Puiseux series of x(t): h(x(t), t) : { xi (s) =b i s v i (1 + O(s)) b i C = C \{}, v i Z t(s) =1 s ω s, t 1 Fractional powers if the winding number ω>1. Observe how the leading exponents v i determine x i (s): v i < : x i (s), v i = : x i (s) b i, v i > : x i (s). Compute v i : log( x i (s 1 ) ) =log( b i )+v i log(s 1 ), log( x i (s 2 ) ) =log( b i )+v i log(s 2 ), extrapolate. Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 9 / 3
Solutions of Initial Forms Substitute x i (s) =b i s v i (1 + O(s)) into f (x) =. f k (x) = a A k c a x a, A k is support of f k. Substitute x i (s) into x a = x a 1 1 x a 2 an 2 xn : n (x i (s)) a i = i=1 n (b i ) a i s a 1v 1 +a 2 v 2 + +a nv n (1 + O(s)). i=1 As s, those monomials that matter are those for which a 1 v 1 + a 2 v 2 + + a n v n = a, v is minimal. in v (f k )= c a x a vanishes at (b 1, b 2,...,b n ). a A k a, v is minimal Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 1 / 3
A Simple Example but a difficult one for basic numerical methods x 2 = f (x, y) = xy = y 2 = Randomization or Embedding: { x 2 + γ 1 y 2 = xy + γ 2 y 2 = or (, ) is an isolated root of multiplicity 3 x 2 + γ 1 z = xy + γ 2 z = y 2 + γ 3 z =, where γ 1, γ 2, γ 3 C are random numbers and z is a slack variable, raises the multiplicity from 3 to 4! Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 11 / 3
Deflation for Isolated Singular Solutions restoring quadratic convergence of Newton s method input: f (x) = a polynomial system; x an approximate solution: f (x ) Stage 1: recondition the problem output: G(x, λ) = an extension to f ; (x, λ ) is a regular solution: G(x, λ )= f (x )=. Stage 2: compute the multiplicity input: g(x) = a good system for x Newton converges quadratically starting from x. Apply algorithms of [Dayton & Zeng, 25] or [Bates, Peterson & Sommese, 26]. output: a quadratically convergent method to refine x and the multiplicity structure of x Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 12 / 3
Deflation Operator Dfl reduces to Corank One Consider f (x) =, N equations in n unknowns, N n. Suppose Rank(A(z )) = R < n for z an isolated zero of f (x) =. Choose h C R+1 and B C n (R+1) at random. Introduce R + 1 new multiplier variables λ =(λ 1, λ 2,...,λ R+1 ). f (x) = Rank(A(x)) = R Dfl(f )(x, λ):= A(x)Bλ = hλ = 1 corank(a(x)b) =1 The operator Dfl is used recursively if necessary, note: (1) # times bounded by multiplicity (2) symbolic implementation easy, but leads to expression swell (3) exploiting the structure for evaluation is efficient Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 13 / 3
The Simple Example with deflation reconditioning the multiple solution x 2 = f (x, y) = xy = y 2 = J f (x, y) = 2x y x 2y z =(, ), m = 3 Rank(J f (z )) = A nontrivial linear combination of the columns of J f (z ) is zero. F(x, y, λ 1 )= 2x y x 2y [ b11 b 21 f (x, y) = ] λ 1 =, random b 11, b 21 h 1 λ 1 = 1, random h 1 C The system F(x, y, λ 1 )= has (,, λ 1 ) as regular zero! Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 14 / 3
Newton s Method with Deflation Input: f (x) = polynomial system; x initial approximation for x ; ɛ tolerance for numerical rank. [A +, R] :=SVD(A(x k ), ɛ); x k+1 := x k A + f (x k ); Gauss-Newton R = #columns(a)? Yes Output: f ; x k+1. No { f (x) = f := Dfl(f )(x, λ) = G(x, λ) = ; Deflation Step λ := LeastSquares(G(x k+1, λ)); k := k + 1; x k := (x k, λ); Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 15 / 3
Multiplicity of an Isolated Zero via Duality Analogy with Univariate Case: z is m-fold zero of f (x) =: f (z )=, f x (z )=, 2 f x 2 (z )=,..., m 1 f x m 1 (z )=. }{{} m = number of linearly independent polynomials annihilating z The dual space D at z is spanned by m linear independent differentiation functionals annihilating z. x 2 1 y 2 = xy Consider again f (x, y) = xy = 1 y 2 = 1x 2 The multiplicity of z =(, ) is 3 because D = span{ [z ], 1 [z ], 1 [z ]}, with ij [z ]= 1 i!j! i+j x i y j f (z ). Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 16 / 3
Computing the Multiplicity Structure following B.H. Dayton and Z. Zeng, ISSAC 25 Looking for differentiation functionals d[z ]= a c a a [z ], with a [z ](p) = Membership criterium for d[z ]: ( 1 a 1!a 2! a n! a 1+a 2 + +a n x a 1 1 x a 2 2 x an n ) p (z ). d[z ] D d[z ](pf i )=, p C[x], i = 1, 2,...,N. To turning this criterium into an algorithm, observe: 1 since d[z ] is linear, restrict p to x k = x k 1 1 x k 2 2 x kn n ; and 2 limit degrees k 1 + k 2 + + k n a 1 + a 2 + + a n, as z = vanishes trivially if not annihilated by a. Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 17 / 3
Computing the Multiplicity Structure An Example f 1 = x 1 x 2 + x 2 1, f 2 = x 1 x 2 + x 2 2 following B.H. Dayton and Z. Zeng f 1 S 1 f 2 x 1 f 1 x 1 f 2 x 2 f 1 S 2 x 2 f 2 x1 2f 1 x1 2f 2 x 1 x 2 f 1 x 1 x 2 f 2 x2 2f 1 a = {}}{ a =1 {}}{ 1 1 a =2 {}}{ 2 11 2 a =3 {}}{ 3 21 12 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 S x2 2 3 f 2 Nullity(S 2 ) = Nullity(S 3 ) stop algorithm D = span{, 1 + 1, 1 + 2 + 11 + 2 } multiplicity = 3 Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 18 / 3
Numerical Irreducible Decomposition specification of input and output Input: f (x) =, x =(x 1, x 2,...,x n ), f =(f 1, f 2,...,f N ) (C[x]) N. The #equations N may differ from n, the #unknowns. A numerical irreducible decomposition consists of 1 for every dimension: a description of sets of solutions, 2 for every pure dimensional solution set: its decomposition in irreducible components. The output is summarized in the following numbers 1 for every dimension: the degree of the solution set, 2 for every pure dimensional solution set: the degrees of all irreducible components, and their multiplicities. Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 19 / 3
Positive Dimensional Solution Sets represented numerically by witness sets Given a system f (x) =, we represent a component of f 1 () of dimension k and degree d by k general hyperplanes L to cut the dimension; and d generic points in f 1 () L. Witness set representations reduce to isolated solutions, with continuation methods we sample solution sets. Using a flag of linear spaces, defined by an decreasing sequence of subsets of the k general hyperplanes, C n L n 1 L 1 L =, we move solutions with nonzero slack values to generic points on lower dimensional components, using a cascade of homotopies. Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 2 / 3
Example of a Homotopy in the Cascade To compute numerical representations of the twisted cubic and the four isolated points, as given by the solution set of one polynomial system, we use the following homotopy: H(x, z 1, t) = (x1 2 x 2)(x 1.5) (x1 3 x 3)(x 2.5) (x 1 x 2 x 3 )(x 3.5) + t γ 1 γ 2 γ 3 t (c + c 1 x 1 + c 2 x 2 + c 3 x 3 ) + z 1 At t = 1: H(x, z 1, t) =E(f )(x, z 1 )=. At t = : H(x, z 1, t) =f (x) =. z 1 = As t goes from 1 to, the hyperplane is removed from the system, and z 1 is forced to zero. Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 21 / 3
Example of a Homotopy in the Cascade To compute numerical representations of the twisted cubic and the four isolated points, as given by the solution set of one polynomial system, we use the following homotopy: H(x, z 1, t) = (x1 2 x 2)(x 1.5) (x1 3 x 3)(x 2.5) (x 1 x 2 x 3 )(x 3.5) + t γ 1 γ 2 γ 3 t (c + c 1 x 1 + c 2 x 2 + c 3 x 3 ) + z 1 At t = 1: H(x, z 1, t) =E(f )(x, z 1 )=. At t = : H(x, z 1, t) =f (x) =. z 1 = As t goes from 1 to, the hyperplane is removed from the system, and z 1 is forced to zero. Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 21 / 3
Computing a Numerical Irreducible Decomposition input: f (x) = a polynomial system with x C n Stage 1: represent the k-dimensional solutions Z k, k =, 1,... output: sequence [W, W 1,...,W n 1 ] of witness sets W k =(E k, E 1 k () \ J k ), deg Z k =#(E 1 k () \ J k ) E k = f + k random hyperplanes, J k = junk Stage 2: decompose Z k, k =, 1,...into irreducible factors output: W k = {W k1, W k2,...,w knk }, k = 1, 2,...,n 1 n k irreducible components of dimension k output: a numerical irreducible decomposition of f 1 () is a sequence of partitioned witness sets Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 22 / 3
The Riemann Surface of z w 3 = : 1.5 1.5 Re(z^1/3) -.5-1 -2-1 Re(z) -1.5-2 -1 Im(z) 1 2 1 2 Loop around the singular point (,) permutes the points. Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 23 / 3
Generating Loops by Homotopies W L represents a k-dimensional solution set of f (x) =, cut out by k random hyperplanes L. Fork other hyperplanes K,wemoveW L to W K, using the homotopy h L,K,α(x, t) =, from t = to1: ( h L,K,α(x, t) = f (x) α(1 t)l(x)+tk (x) ) =, α C. The constant α is chosen at random, to avoid singularities, as t < 1. To turn back we generate another random constant β, and use ( ) h K,L,β (x, t) = f (x) =, β C. β(1 t)k (x)+tl(x) A permutation of points in W L occurs only among points on the same irreducible component. Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 24 / 3
Linear Traces as Stop Criterium Consider f (x, y(x)) = (y y 1 (x))(y y 2 (x))(y y 3 (x)) = y 3 t 1 (x)y 2 + t 2 (x)y t 3 (x) We are interested in the linear trace: t 1 (x) =c 1 x + c. Sample the cubic at x = x and x = x 1. The samples are {(x, y ), (x, y 1 ), (x, y 2 )} and {(x 1, y 1 ), (x 1, y 11 ), (x 1, y 12 )}. Solve { y + y 1 + y 2 = c 1 x + c y 1 + y 11 + y 12 = c 1 x 1 + c to find c, c 1. With t 1 we can predict the sum of the y s for a fixed choice of x. For example, samples at x = x 2 are {(x 2, y 2 ), (x 2, y 21 ), (x 2, y 22 )}. Then, t 1 (x 2 )=c 1 x 2 + c = y 2 + y 21 + y 22. If, then samples come from irreducible curve of degree > 3. Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 25 / 3
Linear Traces an example x x 1 x 2 f 1 () y y1 y2 y1 y11 y21 y2 y12 y22 Use {(x, y ), (x, y 1 ), (x, y 2 )} and {(x 1, y 1 ), (x 1, y 11 ), (x 1, y 12 )} to find the linear trace t 1 (x) =c + c 1 x. At {(x 2, y 2 ), (x 2, y 21 ), (x 2, y 22 )}: c + c 1 x 2 = y 2 + y 21 + y 22? Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 26 / 3
Numerical Primary Decomposition specification of input and output Input: f (x) =, x =(x 1, x 2,...,x n ), f =(f 1, f 2,...,f N ) (C[x]) N. Consider the ideal I generated by the polynomials in f. The output of a numerical primary decomposition is a list of irreducible components: each component is a solution set of an associated prime ideal in the primary decomposition of I, Each component contains an indication whether embedded or isolated isolated means: not contained in another component, its dimension, degree, and multiplicity structure, sufficient information to solve the ideal membership problem. Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 27 / 3
Deflation reveals Embedded Components Anton Leykin, ISSAC 28 Embedded components become visible after deflation. Let I = f 1, f 2,...,f N, k = x k, k = 1, 2,...,n. The first order deflation matrix of I is f 1 1 f 1 2 f 1 n f 1 A (1) I (x) = f 2 1 f 2 2 f 2 n f 2....... f N 1 f N 2 f N n f N Consider A (1) I (x)a T, where a =(a, a 1, a 2,...,a n ) in C[x, a]. Compute a numerical irreducible decomposition to the deflated ideal I (1) = A (1) I (x)a T and apply (x, a) x to the result. Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 28 / 3
An Example Consider I = x 2, xy = x x 2, y. We see the embedded point (, ) =V ( x 2, y ) after one deflation: The deflated ideal is [ A (1) x 2 2x I (x, y) = xy y x ]. I (1) = x 2, xy, a 1 2x, a 1 y + a 2 x. Computing V (I (1) ) via its radical: I (1) = x, ya 1 = x, a 1 x, y. Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 29 / 3
Summary a complexity classification We considered three types of solving: 1 zero dimensional solving typically one single homotopy parallel implementations allow tracking of millions of paths 2 numerical irreducible decomposition need to add extra linear equations different homotopies used after each other 3 numerical primary decomposition one deflation doubles the dimension numerical irreducible decomposition is blackbox Jan Verschelde (UIC) Numerical Algebraic Geometry AMS MRC June 21-27 28 3 / 3