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

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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

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• 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

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• 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.

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• -We can do this using Buchbergers 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
  .

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• 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
• 
  

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• -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.

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• -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

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• -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

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• -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.

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• -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

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• -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

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• -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

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• -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

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• -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 ? first, x second, and so on.

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

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• 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.

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• -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

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• -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!

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• 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

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• -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 !

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• 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.

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• -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.

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• 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)

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• Sources Used
• - Ideals, Varieties, and Algorithms, by Cox,
  Little, OShea
• UTM Springer, 3rd Ed., 2007.
• Thank You!
• See you on Thursday!