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Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Tutorial: Numerical Algebraic Geometry Back to classical algebraic geometry... with more computational power and hybrid symbolic-numerical algorithms Anton Leykin Georgia Tech Waterloo, November 2011

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Outline Homotopy continuation predictor-corrector numerical methods, Newton’s method, (global) homotopy continuation scenarios Singular isolated solutions regularization of singular solutions, deflation, dual spaces/inverse systems Positive dimension witness sets, numerical irreducible decomposition, numerical primary decomposition Certified homotopy tracking numerical zeros, α -theory of Smale, heuristic vs. rigorous path-tracking

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Computational algebraic geometry What is the game? • Level 0: Given a system of polynomial equations in K [ x 1 , ..., x n ] with finitely many solutions, SOLVE . ( K could be Q , Z /p Z , R , C , ... ) • Level 1+: Describe positive-dimensional solutions (curves, surfaces, ...) Classical methods “generalize” linear algebra: • Gröbner basis: a generalization of Gaussian reduction; • Resultant: a generalization of determinant. These methods are symbolic.

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Linear Algebra Numerical Linear Algebra �       � � Algebraic Geometry Numerical Algebraic Geometry �

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Applications Robotics: Stewart-Gough platforms. Griffis-Duffy platform: the solution contains a curve of degree 28. s 2 ℓ 3 ℓ 4 Enumerative algebraic geometry: ℓ 2 solutions of Schubert problems. ✑ ✸ p ✑✑ s 1 ℓ 1 ... control theory, optimization, computer vision, math biology, real algebraic geometry, algebraic curves ...

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Polynomial homotopy continuation • Target system: n equations in n variables, F ( x ) = ( f 1 ( x ) , . . . , f n ( x )) = 0 , where f i ∈ R = C [ x ] = C [ x 1 , ..., x n ] for i = 1 , ..., n . • Start system: n equations in n variables: G ( x ) = ( g 1 ( x ) , . . . , g n ( x )) = 0 , such that it is easy to solve. • Homotopy: for γ ∈ C \ { 0 } consider H ( x , t ) = (1 − t ) G ( x ) + γtF ( x ) , t ∈ [0 , 1] .

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Example target start x 4 1 x 2 + 5 x 2 1 x 3 2 + x 3 x 5 f 1 = 1 − 4 g 1 = 1 − 1 x 2 x 2 f 2 = 1 − x 1 x 2 + x 2 − 8 g 2 = 2 − 1 Start solutions → target solutions: � − 1 ∂H H ( x , t ) = 0 implies d x � ∂H dt = − ∂t . ∂ x

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Example target start − x 2 + 2 x 2 − 1 f = g = The solution of the homotopy equation H ( x, t ) = (1 − t ) g ( x ) + tf ( x ) = (1 − 2 t ) x 2 − 1 + 3 t = 0 is singular for t = 1 / 3 .

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Randomization • Note: the complement of a complex algebraic variety is connected. f g space of polynomials • For all but finite number of γ ∈ C the homotopy H ( x , t ) = (1 − t ) G ( x ) + γtF ( x ) . is regular for 0 ≤ t < 1 .

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Global picture Optimal homotopy: • the continuation paths are regular; • the homotopy establishes a bijection between the start and target solutions. Possible singular scenarios: non-generic diverging paths multiple solutions

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Numerical algebraic geometry • Sommese, Verschelde, and Wampler, Introduction to Numerical AG (2005) • Sommese and Wampler, The numerical solution of systems of polynomials (2005) Software: • PHCpack (Verschelde); • HOM4PS (group of T.Y.Li); • Bertini (group of Sommese); • NAG4M2: Numerical Algebraic Geometry for Macaulay2 (L.). and more, e.g.: Maple’s R OOT F INDING [H OMOTOPY ].

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Possible improvements • Parallel computation: Paths are mutually independent ⇒ linear speedups . • Minimize the number of diverging paths: • Total degree: Number of start solutions = product of degrees of equations (Bézout bound). • Polyhedral homotopies: Number of start solutions = mixed volume of sparse system (BKK bound). • Optimal homotopies: • Cheater’s homotopy; • Special homotopies: e.g., Pieri homotopy .

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Multiple solutions In general, with probability 1, the picture looks like this: Singular end games [Morgan, Sommese, Wampler (1991)]: • power-series method; • Cauchy integral method; • trace method. Deflation: • regularizes an isolated singular solution; • restores quadratic convergence of the Newton’s method. How to describe a singularity?

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Cauchy integral endgame • An implication of Cauchy residue theorem: Let y : U → C holomorphic on a simply connected U ⊂ C , a ∈ C , and C ⊃ C ≃ S 1 be a contour winding I ( C, a ) times around a . Then 1 � y ( z ) y ( a ) = z − a dt, 2 πiI ( C, a ) C • H ( x , t ) = 0 defines (a possibly multivalued function) x = x ( t ) in a neighborhood of t = 1 . • Idea: as the homotopy tracker approaches a singular x ∗ = x (1) use Cauchy integral to compute x ∗ staying away from x ∗ .

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Winding numbers | 1 − t | = ε (1 2 3 4 5)(6 7 8)(9 10)

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Cauchy integral endgame x = x (˜ t ) , a solution to H ( x , t ) = 0 for t = ˜ 1. Pick a point on ˜ t ∈ R ; let ε = 1 − ˜ t . 2. Track the path C = { x (1 − εe iθ ˜ I ) | θ ∈ [0 , 2 π ] } , where ˜ I > 0 is such that x = x (1 − εe iθ ˜ I ) ⇒ θ ∈ { 0 , 2 π } . ˜ 3. Let y ( z ) = x (1 − z ˜ I ) , then y ( z ) is holonomic for | z | < ε (if ε ≪ 1 ). 4. Find numerically the integral x (1) = y (0) = 1 � y ( z ) dz = 1 � x (1 − εe iθ ˜ I ) dθ. 2 π z 2 π | z | = ε [0 , 2 π ] (Note: one may use samples made when tracking the path C .)

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Newton’s method: x ( n +1) = x ( n ) − f ( x ( n ) ) f ′ ( x ( n ) ) Example 1: f ( x ) = x ( x − 1) 3 , x (0) = 0 . 1 x (1) = − 0 . 05000000000000000000000000000000000000000000000000 x (2) = − 0 . 00625000000000000000000000000000000000000000000000 x (3) = − 0 . 00011432926829268292682926829268292682926829268293 x (4) = − 0 . 00000003919561993882928315798471103711494222972094 x (5) = − 0 . 00000000000000460888914457438597268761599543603706 x (6) = − 0 . 00000000000000000000000000006372557744092567103642 Example 2: f ( x ) = x 2 ( x − 1) 3 , x (0) = 0 . 1 x (1) = 0 . 04000000000000000000000000000000000000000000000000 x (2) = 0 . 01866666666666666666666666666666666666666666666667 x (3) = 0 . 00905920745920745920745920745920745920745920745921 x (4) = 0 . 00446662546689373374865785737653016156369492056043 x (5) = 0 . 00221818070337351048684295922675846246257988477728 x (6) = 0 . 00110537952927547542499858913840929687677679537995

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Deflation method Let f ( x ) = ( f 1 ( x ) , ..., f N ( x )) , N ≥ n, f i ( x ) ∈ C [ x ] = C [ x 1 , ..., x n ] . � ∂f i � ∈ C N × n be the Jacobian matrix. Let A ( x ) = ∂x j Given: an approximation x (0) of an exact isolated solution x ∗ , which is singular, i.e., corank A ( x ∗ ) = n − rank A ( x ∗ ) > 0 . Newton’s method in homotopy continuation loses quadratic convergence around x ∗ . Is there a way to restore the convergence? • Want: a symbolic procedure that “makes” x ∗ regular. • Rules: • New variables are allowed. • Assume that the numerical rank of A ( x (0) ) equals A ( x ∗ ) .

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking Deflation step: create an augmented system in C [ x , a ] 1. Introduce n new variables a ; 2. Add equations coming from A ( x ) a = 0 ; Example. Let f 1 = x 3 1 + x 1 x 2 2 , f 2 = x 1 x 2 2 + x 3 2 , f 3 = x 2 1 x 2 + x 1 x 2 2 and x ∗ = 0 . ∂ 1 ∂ 2  3 x 2 1 + x 2  2 x 1 x 2 f 1 2 � � a 1 x 2 2 x 1 x 2 + 3 x 2 f 2 = 0 .   2 2 a 2   2 x 1 x 2 + x 2 x 2 f 3 1 + 2 x 1 x 2 2 3. Compute the rank r of A ( x ∗ ) ; ( r = 0 for our example) 4. Add n − r random linear equations. 5. Find the solution ( x ∗ , a ∗ ) of the augmented system; ( 8 equations) 6. Repeat if ( x ∗ , a ∗ ) is singular. (2 steps for the example) Theorem (L., Verschelde, Zhao) The multiplicity of ( x ∗ , a ∗ ) in the augmented system is smaller than that of x ∗ in the original system.