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TypedGcd

Edit detail for TypedGcd revision 2 of 2

changed:
-  In $$Q(x)[z]$$
  In $$Q(x)[y]$$

GCD and types

Types help to make it clear where the computation happens.

Let's first start with a few abbreviations.

fricas

(1) -> P(R,x)==>UnivariatePolynomial(x,R);

Type: Void

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Z==>Integer;

Type: Void

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Q==>Fraction Z;

Type: Void

Now we compute the gcd in

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p11: P(P(Z, x), y) := 12*x^2*y;

Type: UnivariatePolynomial(y,UnivariatePolynomial(x,Integer))

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p12: P(P(Z, x), y) := 18*x*y^2;

Type: UnivariatePolynomial(y,UnivariatePolynomial(x,Integer))

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gcd(p11, p12)

$\label{eq1}6 \ x \ y$

Type: UnivariatePolynomial(y,UnivariatePolynomial(x,Integer))

Now in

fricas

p21: P(P(Q, x), y) := 12*x^2*y;

Type: UnivariatePolynomial(y,UnivariatePolynomial(x,Fraction(Integer)))

fricas

p22: P(P(Q, x), y) := 18*x*y^2;

Type: UnivariatePolynomial(y,UnivariatePolynomial(x,Fraction(Integer)))

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gcd(p21, p22)

$\label{eq2}x \ y$

Type: UnivariatePolynomial(y,UnivariatePolynomial(x,Fraction(Integer)))

fricas

p31: P(Fraction P(Q, x), y) := 12*x^2*y;

Type: UnivariatePolynomial(y,Fraction(UnivariatePolynomial(x,Fraction(Integer))))

fricas

p32: P(Fraction P(Q, x), y) := 18*x*y^2;

Type: UnivariatePolynomial(y,Fraction(UnivariatePolynomial(x,Fraction(Integer))))

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gcd(p31, p32)

$\label{eq3}y$

Type: UnivariatePolynomial(y,Fraction(UnivariatePolynomial(x,Fraction(Integer))))

And finally in the field

fricas

p41: Fraction P(P(Q, x), y) := 12*x^2*y;

Type: Fraction(UnivariatePolynomial(y,UnivariatePolynomial(x,Fraction(Integer))))

fricas

p42: Fraction P(P(Q, x), y) := 18*x*y^2;

Type: Fraction(UnivariatePolynomial(y,UnivariatePolynomial(x,Fraction(Integer))))

fricas

gcd(p41, p42)

$\label{eq4}1$

Type: Fraction(UnivariatePolynomial(y,UnivariatePolynomial(x,Fraction(Integer))))