Section 2-8 First Applications of Groebner Bases by Pablo Spivakovsky-Gonzalez

1. Section 2-8First Applications of Groebner Basesby Pablo Spivakovsky-Gonzalez We started this chapter with 4 problems: Ideal Description Problem: Does every ideal have a finite generating set? -Yes, solved by Hilbert Basis Theorem in Section 2-5

2. 2. Ideal Membership Problem: Given and an ideal determine if . 3. The Problem of Solving Polynomial Equations: Find all common solutions in of a system of polynomial equations. 4. The Implicitization Problem: Let V be a subset of given parametrically as : : :

3. Find a system of polynomial equations in the that defines the variety. We will now consider how to apply Groebner bases to the 3 remaining problems. The Ideal Membership Problem Combine Groebner bases with the division algorithm, we get the following ideal membership algorithm: given an ideal I, we can decide whether f lies in I as follows. - First, find a Groebner basis for I.

4. -We can do this using Buchberger’s Algorithm from Section 2-7 -Once we have for I, we use Corollary 2 of Section 2-6: Corollary 2 of 2-6: Let be a Groebner basis for an ideal and let . Then if and only if the remainder on division of f by G is 0. -In other words, iff .

5. Example 1 Let and use the grlex order. Let . We want to know if -Step 1: Is the generating set given here a Groebner basis? -No. Recall the precise definition of Groebner basis: Definition: Fix a monomial order. A finite subset of an ideal I is a Groebner basis if

6. -In our case, there are polynomials such as that do not belong to . Therefore, -So the generating set given is not a Groebner basis; we compute one using a computer algebra system (Step 2): -We can now test if our polynomial f is in I.

7. -Step 3: To do this, we divide by G. We obtain -Remainder is 0, so . -Now consider a different case, where We again want to know if . Using our algorithm, we would divide by G as above. -But in this case we can determine by inspection that f does not lie in I, without carrying out the division. -The reason is that is not in the ideal given by

8. -And since G is a Groebner basis, , so if xy does not lie in then f does not lie in I. Solving Polynomial Equations Example 2 -Consider the following system in :

9. -These equations determine -We want to find . -We recall Proposition 9 of Section 2-5: Prop. 9 of 2-5: is an affine variety. In particular, if then . -This implies that we can compute using any basis of I; then let us use a Groebner basis.

10. -We use lex ordering, we get the following basis: -Note that depends on z alone, so we can easily find its roots: -This gives 4 values of z; substituting each of these values back into and gives unique solutions for x and y -We end up with 4 solutions to

11. -By Prop. 9 of 2-5, , so we have found all solutions to the original equations! Example 3 -We wish to find the min. and max. of subject to the constraint . -Applying Lagrange multipliers we obtain the following system:

12. -We begin by computing a Groebner basis for ideal in generated by left-hand sides of the 4 eqns. -We use lex order with -The basis obtained is

13. -This looks terrifying, but notice that the last polynomial depends only on z ! - Setting it equal to 0, we find the following roots: -Now we can substitute each of these values for z into the remaining equations and solve for x and y. We obtain:

14. -Using this we can easily determine the min. and max. values -In Examples 2 and 3 we found Groebner bases for each ideal with respect to lex order. -This gave us eqns. in which variables were successively eliminated. -For our lex ordering, we used -Now notice the order in which variables are eliminated in the Groebner basis: λ first, x second, and so on.

15. -This is not a coincidence! In Chap. 3 we will see why lex order gives a Groebner basis that successively eliminates variables.

16. The Implicitization Problem -Consider the following parametric eqns. : : -Suppose they define a subset of an algebraic variety V in . -How can we find polynomial eqns. in the that define V? -This can be solved by Groebner basis: a complete proof will be given in Chapter 3.

17. -For now, we restrict ourselves to cases in which the are polynomials. -We consider the affine variety in defined by : : -Basic idea: eliminate from the equations. -Once again we try to use Groebner basis to eliminate variables. -We will use lex order in defined by

18. -Suppose we have a Groebner basis of the ideal -We are using lex order, so our Groebner basis should have polynomials that eliminate variables. - are the biggest in our monomial order, so should be eliminated first. -Therefore, Groebner basis for should have some polynomials with only variables -This is what we are looking for!

19. Example 4 -Consider the parametric curve V given by: in . Then let -Now compute Groebner basis using lex order in -We obtain: -Last two polynomials only involve x, y, z

20. -They define a variety of containing V. -By intuition on dimensions (Chap. 1) we can guess that 2 eqns. in define a curve. -Is V the entire intersection of the two surfaces below? -Can there be other curves or surfaces in the intersection? -These questions will be resolved in Chap. 3 !

21. Example 5 -Consider tangent surface of twisted cubic in . -Parametrization of surface: -Compute Groebner basis using lex order with -We obtain a basis G containing 6 elements.

22. -1 element of basis contains only x, y, z terms: -Variety defined by this eqn. is a surface containing the tangent surface to the twisted cubic. -But it is possible that the surface given by the eqn. is strictly bigger than the tangent surface. -This example will be revisited in Chap. 3.

23. Section Summary -Groebner bases combined with division algorithm give complete solution to ideal membership problem. -Groebner bases can be applied to solving polynomial eqns. and implicitization problem. -We used the fact that Groebner bases computed with lex order succeeded in eliminating vars. in a convenient manner -In Chap. 3, we will prove that this always happens! (Elimination Theory)

24. Sources Used - Ideals, Varieties, and Algorithms, by Cox, Little, O’Shea; UTM Springer, 3rd Ed., 2007. Thank You! See you on Thursday!