## Chapter 39Generating Universal Gröbner Basis- MAY NOT WORK - untested in 1999

These commands are useful for generating a universal GB. If one starts with a GB with respect to any order this can be used to impose equivalence classes on the set of all order s. The commands below characterize these equivalence classes and select representatives from each of them. See [HSH]. this selects

#### 39.0.2 AllOrders[aListofPolynomials, aListofIndeterminants]

Aliases: None
Description: AllOrders[aListOfPolynomials, aListOfIndeterminants] returns A list of graded lexicographical orders of the given indeterminants. Each order, represented as a list, is a member of an equivalence class of orders that produce the same leading terms in each of the polynomials in the list.
Arguments: aListOfPolynomials is a list of polynomials. aListofIndeterminants is a list of noncommutative indeterminants.
Comments / Limitations: Not available before NCAlgebra 1.2

#### 39.0.3 EquivalenceClasses[aListOfPolynomials] or EquivalenceClasses[aListOfPolynomials,Simpler]

Aliases: None
Description: EquivalenceClasses[aListOfPolynomials] returns a logical expression that represents the equivalence classes of orders that produce the same leading terms in each of the polynomials in the list. If Simpler is used and it is False, no additional processing is done. If Simpler is used and it is True, the expression will be simplified as much as possible. In this case, the EquivalenceClasses command will take longer than if there were no second argument.
Arguments: aListOfPolynomials is a list of polynomials. Simpler is either True or False.
Comments / Limitations: Not available before NCAlgebra 1.2

#### 39.0.4 UniversalBasis[aListOfPolynomials, NumberOfIterations]

Aliases: None
Description: UniversalBasis[aListOfPolynomials, NumberOfIterations] finds a universal Gröbner Basis with respect to the set of all graded lexicographical orders on the indeterminants in aListOfPolynomials for the ideal generated by aListOfPolynomials. NumberOfIterations is passed as the second argument to NCMakeRules.
Arguments: aListOfPolynomials is a list of polynomials. numberOfIterations is a positive integer.
Comments / Limitations: Not available before NCAlgebra 1.2

### 39.1 Very Technical Commands

#### 39.1.1 GroebnerCutOffFlag[n_Integer]

Aliases: None
Description: Turns the “cutting off” operations either on or off.
Arguments: n is an natural number.
Comments / Limitations: Not available before NCAlgebra 1.2

#### 39.1.2 GroebnerCutOffMin[n_Integer]

Aliases: None
Description: Sets the min for sum on polynomial degree cut offs.
Arguments: n is an natural number.
Comments / Limitations: Not available before NCAlgebra 1.2

#### 39.1.3 GroebnerCutOffSum[n_Integer]

Aliases: None
Description: Sets the min for sum on polynomial degree cut offs.
Arguments: n is an natural number.
Comments / Limitations: Not available before NCAlgebra 1.2