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TypeTowerDemo

Edit detail for TypeTowerDemo revision 4 of 5

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Editor: hemmecke Time: 2008/08/18 12:20:55 GMT-7
Note:

changed:
-there is no applicaple operation.
there is no applicable operation.

This page demonstrates some features of Axiom.

Let's begin with the construction of a polynomial ring in the indeterminate with coefficients from the ring of square matrices with entries that are polynomials where is the Galois field with 3 elements.

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F := PrimeField 3

$\label{eq1}\hbox{\axiomType{PrimeField}\ } (3)$

(1)

Type: Type

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P := UnivariatePolynomial(x, F)

$\label{eq2}\hbox{\axiomType{UnivariatePolynomial}\ } (x , \hbox{\axiomType{PrimeField}\ } (3))$

(2)

Type: Type

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S := SquareMatrix(2, P)

$\label{eq3}\hbox{\axiomType{SquareMatrix}\ } (2, \hbox{\axiomType{UnivariatePolynomial}\ } (x , \hbox{\axiomType{PrimeField}\ } (3)))$

(3)

Type: Type

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R := UnivariatePolynomial(z, S)

$\label{eq4}\hbox{\axiomType{UnivariatePolynomial}\ } (z , \hbox{\axiomType{SquareMatrix}\ } (2, \hbox{\axiomType{UnivariatePolynomial}\ } (x , \hbox{\axiomType{PrimeField}\ } (3))))$

(4)

Type: Type

OK, now we have the type . Let's construct an element. We start with constructing some coefficients first.

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s1:S := matrix[[2*x +1 ,x^2-1],[0,x-1]]

$\label{eq5}\left[ \begin{array}{cc} {{2 \ x}+ 1}&{{{x}^{2}}+ 2} \ 0 &{x + 2}$

(5)

Type: SquareMatrix?(2,UnivariatePolynomial?(x,PrimeField?(3)))

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s2 := transpose s1

$\label{eq6}\left[ \begin{array}{cc} {{2 \ x}+ 1}& 0 \ {{{x}^{2}}+ 2}&{x + 2}$

(6)

Type: SquareMatrix?(2,UnivariatePolynomial?(x,PrimeField?(3)))

And now we build the polynomial.

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r: R := z^2 + s1*z + s2

$\label{eq7}\begin{array}{@{}l} \displaystyle {{z}^{2}}+{{\left[ \begin{array}{cc} {{2 \ x}+ 1}&{{{x}^{2}}+ 2} \ 0 &{x + 2}$

(7)

Type: UnivariatePolynomial?(z,SquareMatrix?(2,UnivariatePolynomial?(x,PrimeField?(3))))

Of course, since we work in characteristic 3, the following sum must be zero.

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r+ 2*r

$\label{eq8}0$

(8)

Type: UnivariatePolynomial?(z,SquareMatrix?(2,UnivariatePolynomial?(x,PrimeField?(3))))

Note that this is not the integer 0, but it is still a polynomial of type .

Asking for the degree of is no problem, because is a univariate polynomial ring.

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degree r

$\label{eq9}2$

(9)

Type: PositiveInteger?

Let us add 1 to .

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r+1

$\label{eq10}\begin{array}{@{}l} \displaystyle {{z}^{2}}+{{\left[ \begin{array}{cc} {{2 \ x}+ 1}&{{{x}^{2}}+ 2} \ 0 &{x + 2}$

(10)

Type: UnivariatePolynomial?(z,SquareMatrix?(2,UnivariatePolynomial?(x,PrimeField?(3))))

Oh, interesting, Axiom figured out that by 1 we actually meant the polynomial 1 in .

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one: R := 1

$\label{eq11}\left[ \begin{array}{cc} 1 & 0 \ 0 & 1$

(11)

Type: UnivariatePolynomial?(z,SquareMatrix?(2,UnivariatePolynomial?(x,PrimeField?(3))))

No no, it is not the unit square matrix. It is the unit square matrix multiplied by . Look at the type.

So let's see what happens if we multiply by itself.

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r2 := r*r

$\label{eq12}\begin{array}{@{}l} \displaystyle {{z}^{4}}+{{\left[ \begin{array}{cc} {x + 2}&{{2 \ {{x}^{2}}}+ 1} \ 0 &{{2 \ x}+ 1}$

(12)

Type: UnivariatePolynomial?(z,SquareMatrix?(2,UnivariatePolynomial?(x,PrimeField?(3))))

Well, of course there is a common factor of an . Can Axiom find it?

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gcd(r2, r)

   There are 4 exposed and 3 unexposed library operations named gcd 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                               )display op gcd
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named gcd 
      with argument type(s) 
UnivariatePolynomial(z,SquareMatrix(2,UnivariatePolynomial(x,PrimeField(3))))
UnivariatePolynomial(z,SquareMatrix(2,UnivariatePolynomial(x,PrimeField(3))))

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Ooops. What does that say?:

  Cannot find a definition or applicable library operation named gcd.

Ah, of course, the coefficient ring of is the matrix ring and this is unfortunately not a field (not even an integral domain), so we cannot simply apply Euclid's algorithm. Axiom simply stops by telling you that there is no applicable operation.

Of course, Axiom can compute a gcd of univariate polynomials.

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