19.6 Operations for Polynomials

The following operations are always available for polynomials. The operands must have a common base ring, no implicit conversions are performed.

f + g

The operator + evaluates to the sum of the polynomials f and g, which must be polynomials over a common base ring.

    gap> f := Polynomial( GF(2), [Z(2), Z(2)] );
    Z(2)^0*(X(GF(2)) + 1)
    gap> f + f;
    0*X(GF(2))^0
    gap> g := Polynomial( GF(4), [Z(2), Z(2)] );
    X(GF(2^2)) + Z(2)^0
    gap> f + g;
    Error, polynomials must have the same ring

f + scl
scl + f

The operator + evaluates to the sum of the polynomial f and the scalar scl, which must lie in the base ring of f.

    gap> x := Indeterminate( Integers );; x.name := "x";;
    gap> h := Polynomial( Integers, [1,2,3,4] );
    4*x^3 + 3*x^2 + 2*x + 1
    gap> h + 1;
    4*x^3 + 3*x^2 + 2*x + 2
    gap> 1/2 + h;
    Error, <l> must lie in the base ring of <r>

f - g

The operator - evaluates to the difference of the polynomials f and g, which must be polynomials over a common base ring.

    gap> x := Indeterminate( Integers );; x.name := "x";;
    gap> h := Polynomial( Integers, [1,2,3,4] );
    4*x^3 + 3*x^2 + 2*x + 1
    gap> h - 2*h;
    -4*x^3 - 3*x^2 - 2*x - 1

f - scl
scl - f

The operator - evaluates to the difference of the polynomial f and the scalar scl, which must lie in the base ring of f.

    gap> x := Indeterminate( Integers );; x.name := "x";;
    gap> h := Polynomial( Integers, [1,2,3,4] );
    4*x^3 + 3*x^2 + 2*x + 1
    gap> h - 1;
    4*x^3 + 3*x^2 + 2*x
    gap> 1 - h;
    -4*x^3 - 3*x^2 - 2*x

f * g

The operator * evaluates to the product of the two polynomials f and g, which must be polynomial over a common base ring.

    gap> x := Indeterminate(Integers);; x.name := "x";;
    gap> h := 4*x^3 + 3*x^2 + 2*x + 1;
    4*x^3 + 3*x^2 + 2*x + 1
    gap> h * h;
    16*x^6 + 24*x^5 + 25*x^4 + 20*x^3 + 10*x^2 + 4*x + 1

f * scl
scl * f

The operator * evaluates to the product of the polynomial f and the scalar scl, which must lie in the base ring of f.

    gap> f := Polynomial( GF(2), [Z(2), Z(2)] );
    Z(2)^0*(X(GF(2)) + 1)
    gap> f - Z(2);
    X(GF(2))
    gap> Z(4) - f;
    Error, <l> must lie in the base ring of <r>

f ^ n

The operator ^ evaluates the the n-th power of the polynomial f. If n is negative ^ will try to invert f in the Laurent polynomial ring ring.

    gap> x := Indeterminate(Integers);; x.name := "x";;
    gap> k := x - 1 + x^-1;
    x - 1 + x^(-1)
    gap> k ^ 3;
    x^3 - 3*x^2 + 6*x - 7 + 6*x^(-1) - 3*x^(-2) + x^(-3)
    gap> k^-1;
    Error, cannot invert <l> in the laurent polynomial ring

f / scl

The operator / evaluates to the product of the polynomial f and the inverse of the scalar scl, which must be invertable in its default ring.

    gap> x := Indeterminate(Integers);; x.name := "x";;
    gap> h := 4*x^3 + 3*x^2 + 2*x + 1;
    4*x^3 + 3*x^2 + 2*x + 1
    gap> h / 3;
    (4/3)*x^3 + x^2 + (2/3)*x + (1/3) 

scl / f

The operator / evaluates to the product of the scalar scl and the inverse of the polynomial f, which must be invertable in its Laurent ring.

    gap> x := Indeterminate(Integers);; x.name := "x";;
    gap> 30 / x;
    30*x^(-1)
    gap> 3 / (1+x);
    Error, cannot invert <l> in the laurent polynomial ring 

f / g

The operator / evaluates to the quotient of the two polynomials f and g, if such quotient exists in the Laurent polynomial ring. Otherwise / signals an error.

    gap> x := Indeterminate(Integers);; x.name := "x";;
    gap> f := (1+x+x^2) * (3-x-2*x^2);
    -2*x^4 - 3*x^3 + 2*x + 3
    gap> f / (1+x+x^2);
    -2*x^2 - x + 3
    gap> f / (1+x);
    Error, cannot divide <l> by <r> 

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