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GCD and types

Types help to make it clear where the computation happens.

Let's first start with a few abbreviations.

fricas
P(R,x)==>UnivariatePolynomial(x,R);
Type: Void
fricas
Z==>Integer;
Type: Void
fricas
Q==>Fraction Z;
Type: Void

Now we compute the gcd in

Z[x][y] 

fricas
p11: P(P(Z, x), y) := 12*x^2*y;
fricas
p12: P(P(Z, x), y) := 18*x*y^2;
fricas
gcd(p11, p12)

\label{eq1}6 \  x \  y(1)

Now in

Q[x][y] 

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)))
fricas
gcd(p21, p22)

\label{eq2}x \  y(2)
Type: UnivariatePolynomial(y,UnivariatePolynomial(x,Fraction(Integer)))

In

Q(x)[y] 

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))))
fricas
gcd(p31, p32)

\label{eq3}y(3)
Type: UnivariatePolynomial(y,Fraction(UnivariatePolynomial(x,Fraction(Integer))))

And finally in the field

Q(x)(y) 

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(4)
Type: Fraction(UnivariatePolynomial(y,UnivariatePolynomial(x,Fraction(Integer))))




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