Describing algebraic varieties to a computer

Transcription

1 Describing algebraic varieties to a computer Jose Israel Rodriguez University of Chicago String Pheno 2017 July 6, 2017

2 Outline Degree of a curve Component membership test Monodromy and trace test Witness sets References: Multiprojective witness sets and a trace test Jonathan Hauenstein and [-] Trace test Anton Leykin, [-], Frank Sottile The maximum likelihood degree of mixtures of independence models [-] and Botong Wang (Counting critical points)

3 Testing membership Intersect a curve V with a hyperplane L. Consider a curve V C n (may be reducible). [Draw] For a general hyperplane L of C n, [Draw] the degree of V is the cardinality of L V. Question (Membership): Given a point B C n, is B in the V? If we know that equations F = 0 define V, then check F (B) = 0. Membership test: Let M denote a general hyperplane through B. Deform L M to determine M V. Check if B M V. If B is not in M V then B is not in V by definition of degree. [Draw] But what about restricting to components (lacking equations)???? Problem: How to sort the points V L by irreducible component?

4 Testing membership Intersect a curve V with a hyperplane L. Consider a curve V C n (may be reducible). [Draw] For a general hyperplane L of C n, [Draw] the degree of V is the cardinality of L V. Question (Membership): Given a point B C n, is B in the V? If we know that equations F = 0 define V, then check F (B) = 0. Membership test: Let M denote a general hyperplane through B. Deform L M to determine M V. Check if B M V. If B is not in M V then B is not in V by definition of degree. [Draw] But what about restricting to components (lacking equations)???? Problem: How to sort the points V L by irreducible component?

5 Testing membership Intersect a curve V with a hyperplane L. Consider a curve V C n (may be reducible). [Draw] For a general hyperplane L of C n, [Draw] the degree of V is the cardinality of L V. Question (Membership): Given a point B C n, is B in the V? If we know that equations F = 0 define V, then check F (B) = 0. Membership test: Let M denote a general hyperplane through B. Deform L M to determine M V. Check if B M V. If B is not in M V then B is not in V by definition of degree. [Draw] But what about restricting to components (lacking equations)???? Problem: How to sort the points V L by irreducible component?

6 Testing membership Intersect a curve V with a hyperplane L. Consider a curve V C n (may be reducible). [Draw] For a general hyperplane L of C n, [Draw] the degree of V is the cardinality of L V. Question (Membership): Given a point B C n, is B in the V? If we know that equations F = 0 define V, then check F (B) = 0. Membership test: Let M denote a general hyperplane through B. Deform L M to determine M V. Check if B M V. If B is not in M V then B is not in V by definition of degree. [Draw] But what about restricting to components (lacking equations)???? Problem: How to sort the points V L by irreducible component?

7 Testing membership Intersect a curve V with a hyperplane L. Consider a curve V C n (may be reducible). [Draw] For a general hyperplane L of C n, [Draw] the degree of V is the cardinality of L V. Question (Membership): Given a point B C n, is B in the V? If we know that equations F = 0 define V, then check F (B) = 0. Membership test: Let M denote a general hyperplane through B. Deform L M to determine M V. Check if B M V. If B is not in M V then B is not in V by definition of degree. [Draw] But what about restricting to components (lacking equations)???? Problem: How to sort the points V L by irreducible component?

8 Testing membership Intersect a curve V with a hyperplane L. Consider a curve V C n (may be reducible). [Draw] For a general hyperplane L of C n, [Draw] the degree of V is the cardinality of L V. Question (Membership): Given a point B C n, is B in the V? If we know that equations F = 0 define V, then check F (B) = 0. Membership test: Let M denote a general hyperplane through B. Deform L M to determine M V. Check if B M V. If B is not in M V then B is not in V by definition of degree. [Draw] But what about restricting to components (lacking equations)???? Problem: How to sort the points V L by irreducible component?

9 Testing membership Intersect a curve V with a hyperplane L. Consider a curve V C n (may be reducible). [Draw] For a general hyperplane L of C n, [Draw] the degree of V is the cardinality of L V. Question (Membership): Given a point B C n, is B in the V? If we know that equations F = 0 define V, then check F (B) = 0. Membership test: Let M denote a general hyperplane through B. Deform L M to determine M V. Check if B M V. If B is not in M V then B is not in V by definition of degree. [Draw] But what about restricting to components (lacking equations)???? Problem: How to sort the points V L by irreducible component?

10 Monodromy Deforming L to itself is a monodromy used to sort the points in V L. Repeatedly deform L to itself to find points in L V that are on the same component. In example C s go to C s and D s go to D s. Illustrate on circle (Short move then rotation). Problem: No stopping criteria to say that we are done sorting. (Reminder: we are working over the complex numbers.) Use the trace (coordinate-wise average) of a set of points.

11 Monodromy Deforming L to itself is a monodromy used to sort the points in V L. Repeatedly deform L to itself to find points in L V that are on the same component. In example C s go to C s and D s go to D s. Illustrate on circle (Short move then rotation). Problem: No stopping criteria to say that we are done sorting. (Reminder: we are working over the complex numbers.) Use the trace (coordinate-wise average) of a set of points.

12 Trace The trace of a set of points S is the coordinate-wise average. Let L t denote a family of hyperplanes defined by H + t = 0. Let X be an irreducible component of V C n. Suppose S X L t. [Trace test] The trace of S is affine linear in t if and only if S = X L t. Numerical irreducible decomposition using projections from points on the components by A.J. Sommese, J. Verschelde, and C.W. Wampler. [Draw to illustrate affine linear in t] Not limited to curves. Take L to be defined by linear equations H 1,H 2,...,H dimv.

13 Trace The trace of a set of points S is the coordinate-wise average. Let L t denote a family of hyperplanes defined by H + t = 0. Let X be an irreducible component of V C n. Suppose S X L t. [Trace test] The trace of S is affine linear in t if and only if S = X L t. Numerical irreducible decomposition using projections from points on the components by A.J. Sommese, J. Verschelde, and C.W. Wampler. [Draw to illustrate affine linear in t] Not limited to curves. Take L to be defined by linear equations H 1,H 2,...,H dimv.

14 Trace The trace of a set of points S is the coordinate-wise average. Let L t denote a family of hyperplanes defined by H + t = 0. Let X be an irreducible component of V C n. Suppose S X L t. [Trace test] The trace of S is affine linear in t if and only if S = X L t. Numerical irreducible decomposition using projections from points on the components by A.J. Sommese, J. Verschelde, and C.W. Wampler. [Draw to illustrate affine linear in t] Not limited to curves. Take L to be defined by linear equations H 1,H 2,...,H dimv.

15 Trace The trace of a set of points S is the coordinate-wise average. Let L t denote a family of hyperplanes defined by H + t = 0. Let X be an irreducible component of V C n. Suppose S X L t. [Trace test] The trace of S is affine linear in t if and only if S = X L t. Numerical irreducible decomposition using projections from points on the components by A.J. Sommese, J. Verschelde, and C.W. Wampler. [Draw to illustrate affine linear in t] Not limited to curves. Take L to be defined by linear equations H 1,H 2,...,H dimv.

16 Component membership test Intersect a curve V with a hyperplane L with the intersection sorted by component. Consider a curve V C n with irreducible component X. Component membership test: Let M denote a general hyperplane through B and suppose we have L X (This can be obtained by monodromy if given a point). Deform L M to determine M X from L X. Check if B M X. If not then B is not in X by definition of degree. [Draw]

17 Component membership test Intersect a curve V with a hyperplane L with the intersection sorted by component. Consider a curve V C n with irreducible component X. Component membership test: Let M denote a general hyperplane through B and suppose we have L X (This can be obtained by monodromy if given a point). Deform L M to determine M X from L X. Check if B M X. If not then B is not in X by definition of degree. [Draw]

18 Component membership test Intersect a curve V with a hyperplane L with the intersection sorted by component. Consider a curve V C n with irreducible component X. Component membership test: Let M denote a general hyperplane through B and suppose we have L X (This can be obtained by monodromy if given a point). Deform L M to determine M X from L X. Check if B M X. If not then B is not in X by definition of degree. [Draw]

19 Witness sets We use witness sets to describe algebraic varieties to a computer. A witness set for a d dimensional algebraic variety X in C n is a triple of information: 1. Equations F = 0 that define a variety containing X as an irreducible component. 2. Linear equations H 1 = 0,...,H d = 0 defining a general codimension d linear space L. 3. Witness points: Numerical approximations to the points in X L. Advantages: Sample points. Test membership. Refine approximations. Computational Savings (two line summary): Monodromy homotopy for partial information Decomposition for focus.

20 Systems with groups of indeterminants Many systems define varieties naturally in C n 1 C n 2 C n k. Our results: develop a trace test for these varieties. Euclidean distance degree C n x C n u. Method of moments C n µ,σ C r a C n m. Found 248, 400 solutions with monodromy (special Galois group). Alt s problem in kinematics C 1 C 12. Tensor completion: 21,288,960 solutions. Likelihood equations: C n p C n u. Optimization: primal variables and Lagrange multipliers C n x C m λ.

21 How do we find L V? Regeneration is an equation by equation technique for solving systems of equations. Consider F := x 2 1 x 2 H := 2x 1 + 3x F = 0 defines a curve V. H defines L. We begin with linear algebra and regenerate to nonlinear systems. [Draw.]

22 How do we find L V? Regeneration is an equation by equation technique for solving systems of equations. Consider F := x 2 1 x 2 H := 2x 1 + 3x F = 0 defines a curve V. H defines L. We begin with linear algebra and regenerate to nonlinear systems. [Draw.]

23 How do we find L V? Regeneration is an equation by equation technique for solving systems of equations. Consider F := x 2 1 x 2 H := 2x 1 + 3x F = 0 defines a curve V. H defines L. We begin with linear algebra and regenerate to nonlinear systems. [Draw.]

24 Our results Multiprojective witness sets and a trace test We introduce witness sets for multiprojective varieties A trace test for multiprojective varieties. A membership test for multiprojective varieties Trace test Twelve pages! We have a dimension reduction for C n C m to C 1 C 1 Contact: Jose Israel Rodriguez JoIsRo@Uchicago.edu