1. Determination of dimension of variety & some applications Author: Wenyuan Wu Supervisor: Greg Reid

2. Organization of this talk 1. Brief introduction to Numerical Algebra Geometry 2. Definition of dimension of algebraic varieties 3. A new way to construct square system for PHC 4. Algorithm of highestdim to compute the dimension of varieties Theorem about dimension Drop the dimension of subsystem by Regular sequence 5. Some applications of highestdim 6. Conclusion

3. Some Methods in Computational Algebra Geometry Grobner BasisSymbolictransform the equations IdealslowAll the information of Ideal Triangular decomposition Symbolictransform the equations Variety / radical Ideal fastAll the information of Variety Homotopy Method hybridtransform the solutions Generic points fasterApprox. information Variety The basic problem of algebraic geometry is to compute ideals and varieties. The most important invariant of variety in affine space is its dimension. GB and Triangular decomposition can compute it, but it is not reasonable to get dimension by computing all the info. Homotopy Method is key point of numerical algebra geometry, a new area developed by Andrew Sommese, Jan Verschelde and Charles Wampler. It gives us a cheaper way to compute “integer or boolean” result of varieties under a given tolerance.

4. Numerical Algebra Geometry Homotopy method is a global method which can compute all the isolated roots. Then we can use Newton method to refine these roots up to arbitrary precision. Theorem 1. All isolated complex solutions of a square polynomial system can be computed by homotopy method [11] [12]. PHC package PHC, a general-purpose solver for polynomial systems by homotopy continuation is a very powerful and efficient software, written by Professor Jan Verschelde, University of Illinois at Chicago. It underlies all the algorithms in the paper. Witness set [?] new representation is the key idea to describe positive dimension varieties.

5. Dimension of Variety Algebraic definition The degree of the Hilbert polynomial HP R/I is the dimension of the variety V(I). [1] Geometric definition The dimension of V at a point P, written dim P V is n – min U (rank(J(V)), U runs over all points of V arbitrarily close to P[2]. The dimension of V, written dimV, is max P in V dim P V. By convention dim = -1 Computing the dimension of variety by slicing the variety by random planes [3] Theorem 2. Given a polynomial system, dimV = k is equivalent to, with probability one, and Proof: dimV= n – min(rank(J(V)))= k min(rank(J(V))) = n – k min(rank(J(V+L))) = n dim(V+L)=0 dimV < k+1 and dim(V+L) 0 min[rank(J(V+L))] n min[rank(J(V))] n-k n-min[rank(J(V))] k dimV k dimV = k

6. Techniques for solving polynomials by PHC Decrease the Bezout Bound (BKK bound) of system complexity of PHC depends on number of paths split system into cases sort polynomials by total degree choose subsystem Decrease the number of variables track each path, it needs to solve Ax=b, require n^3 arithmetic flops solve linear subsystem first then eliminate variables by substitution Construct square system for PHC (next page)

7. Building square system Because PHC only process square polynomial system and many input systems are not square, we need to transform them into square ones. There are two ways to do it: 1. Embedding method (old), introduce some extra variables then project the solutions into original space [8]. with larger Bezout Bound 2. Truncating method (new), only choose a part of system. with smaller Bezout Bound We choose this method because of its lower complexity. The system often is overdetermined system during computing the dimension of variety by slicing method.

8. Problems and solutions 1. How to solve the original system by choosing subsystem ? solve the subsystem by PHC then substitute the points into other polynomials to check if these points satisfy all the other polynomials. 2. How to check a points satisfies a polynomial ? tolerance 3. How to choose 0 dimension subsystem from original system ? if then = comroots(,), Example : comroots(, ) =, but how to guarantee the dimension must drop by one at each step?

9. Regular/non-regular equations Definition 1. If, can’t drop the dimension, then we call a “non-regular” equation w.r.t.. Otherwise we call a “regular” equation w.r.t. Definition 2. Given a polynomials sequence, if is regular w.r.t., then we call this sequence is regular sequence. Theorem 2. Given a consistent polynomial system, if then. [4] Proposition 2. If is regular sequence and, then Proposition 3. Given a regular sequence the following two statements are equivalent: 1. is a “non-regular” equation w.r.t. 2. comroots(, )

10. How to build the regular sequence Common roots table 12……N+ 11…02 01 01 … 00 00 00 0 Assume {N isolated points} product of this column Col(j) This is a N+1 by m-n+k +1 matrix, 0 j th point does not satisfy, element at (i,j) = 1 j th point satisfy, * Sum of this row Row(i)

11. continue Case 1 = 0, is regular Row(i) > 0, is not regular If there is i, Row(i) =0 then we find a regular polynomial of the sequence and we can use it to drop the dimension of the subsystem. Case 2 0, is not the com root. Col (j) = 1, is a com root If there is Col(j) = 1, is a common root, and dimV = k; If no Row(i)=0, no Col(j)=1, then we need to go to Case 3.

12. continue Case 3 let are random numbers. then Proposition 4. must be a regular equation w.r.t with probability 1. Proof: Let us consider the values of at ith point of as a vector p. It should be a nonzero vector, otherwise the ith element must be “ 1 ” in last row. at each point is not zero,otherwise the random vector r must be perpendicular with p. The probability is 0.

13. Algorithm of highestdim input : sys = output: for k from n-1 to max(n-m,0) do NLeqn:= ; Rest:= ; Leqn:= k random planes; newsys:= Nleqn Leqn; sol:= PHC(newsys); if sol= then return –1; end if; com_root_table:= create_table(sol,rest); if there is a “1” in last row then return k; else if there is one “0” in the last column at s row then swap ( ); else insert between end if; end do;

14. Applications Union of varieties Intersection of varieties Difference of two varieties case 1. if then case 2. if then case 3. if then Projection of variety Proposition 7. Given a polynomial system, if the expand system, k+1<n, n is number of variables, has no solution and has solution then are random planes in space. Definition 3. A projection (elimination) of variety V in U plane, is: := { eliminate all the values of at p if : p V}

15. continue Radical Ideal Membership Test Let be polynomials and let y be an extra variable. Let I be the ideal generated by. Then we have Inclusion/equality of varieties for all the i and Radical Ideal Test

16. Conclusion a new numerical symbolic method “highestdim” which has lower complexity to determine the dimension of variety a new way to test radical ideal membership based on “highestdim” Application in RIF package of Maple

17. Reference [2] David Cox, John Little and Donal O’Shea. Using Algebra Geomery. Springer- Verlag, (1998) New York. [4] David Cox, John Little and Donal O’Shea. Ideals, Varieties, and Algorithms. Springer-Verlag, (1997) New York.Ideals, Varieties, and Algorithms. [8] Andrew J. Sommese and Jan Verschelde: Numerical Homotopies to compute Generic Points on Positive Dimensional Algebraic Sets. The Abstract and gzipped postscript file, Revised version. Journal of Complexity 16(3):572-602, 2000.Andrew J. SommeseAbstract gzipped postscript file, Revised version. [11] GARCIA, C. B. and ZANGWILL, W. I., "Finding all solutions to polynomial systems and other systems of equations," Math. Program. 16, pp. 159-176 (1979). [12] T.Y. Li and X. Wang. Nonlinear homotopies for solving deficient polynomial systems with parameters. SIAM J. Numer. Anal., 29(4):1104-1118, 1992. Keith Kendig, Elementary Algebraic Geometry. Springer-Verlag, (1997) New York.