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Section 2-8First Applications of Groebner Bases
by Pablo Spivakovsky-Gonzalez
We started this chapter with 4 problems:
1. Ideal Description Problem: Does every ideal
have a finite generating set?
-Yes, solved by Hilbert Basis Theorem in
Section 2-5
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
:
: :
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 ProblemCombine 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.
-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 .
Example 1Let
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
-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.
-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
-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 :
-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.
-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
-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:
-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
-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:
-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.
-This is not a coincidence! In Chap. 3 we will see why lex
order gives a Groebner basis that successively eliminates
variables.
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.
-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
-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!
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
-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 !
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.
-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.
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)
Sources Used- Ideals, Varieties, and Algorithms, by Cox, Little, O’Shea;
UTM Springer, 3rd Ed., 2007.
Thank You!
See you on Thursday!
