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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, and a power-product is 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 P ::= QQ[x,y,z];
/**/ f := 3*x*y*z + x*y^2;
/**/ f;
x*y^2 + 3*x*y*z
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/**/ use P ::= 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 +x*y;
/**/ G := x;
/**/ F/G;
x + y
-- /**/ F/(x+z); --> !!! ERROR !!! as expected: Inexact division
/**/ F^2;
x^4 +2*x^3*y +x^2*y^2
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