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2.11.1 Introduction to RINGELEM
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An object of type RINGELEM in CoCoA represents an element of a ring.
To fix terminology about polynomials (elements of a polynomial ring):
a polynomial is a sum of terms;
each term is the product of a coefficient and power-product,
a power-product being a product of powers of indeterminates.
(In English it is standard to use
monomial to mean a power-product,
however, in other languages, such as Italian, monomial connotes a
power-product multiplied by a scalar.
In the interest of world peace, we will use the term power-product in those
cases where confusion may arise.)
/**/ Use R ::= QQ[x,y,z];
/**/ F := ***3xyz + xy^2***;
/**/ F;
xy^2 + 3xyz
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Use R ::= QQ[x[1..5]];
Sum([x[N]^2 | N In 1..5]);
x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2 + x[5]^2
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CoCoA always keeps polynomials ordered with respect to the
term-orderings of their corresponding rings.
The following algebraic operations on polynomials are supported:
F^N, +F, -F, F*G, F/G if G divides F, F+G, F-G,
where F, G are polynomials and N is an integer. The result may be a
rational function.
Use R ::= QQ[x,y,z];
F := x^2+xy;
G := x;
F/G;
x + y
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F/(x+z);
(x^2 + xy)/(x + z)
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F^2;
x^4 + 2x^3y + x^2y^2
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F^(-1);
1/(x^2 + xy)
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