Asir: 9.2 Operations over algebraic numbers

In the previous section, we explained about the representation of algebraic numbers and their defining method. Here, we describe operations on algebraic numbers. Only a few functions are built-in, and almost all functions are provided as user defined functions. The file containing their definitions is ‘sp’, and it is placed under the same directory as ‘gr’ is placed, i.e., the standard library directory of Asir.

[0] load("gr")$
[1] load("sp")$

Or if you always need them, it is more convenient to include the load commands in ‘$HOME/.asirrc’.

Like the other numbers, algebraic numbers can get arithmetic operations applied. Simplification, however, by defining polynomials are not automatically performed. It is left to users to simplify their expressions. A fatal error shall result if the denominator expression will be simplified to 0. Therefore, be careful enough when you will create and deal with algebraic numbers which may denominators in their expressions.

Use simpalg() function for simplification of algebraic numbers by defining polynomials.

[49] T=A0^2+1;
(#0^2+1)
[50] simpalg(T);
0

Function simpalg() simplifies algebraic numbers which have the same structures as rational expressions in their appearances.

[39] A0=newalg(x^2+1);      
(#0)
[40] T=(A0^2+A0+1)/(A0+3);
((#0^2+#0+1)/(#0+3))
[41] simpalg(T);
(3/10*#0+1/10)
[42] T=1/(A0^2+1);
((1)/(#0^2+1))
[43] simpalg(T);
div : division by 0
stopped in invalgp at line 258 in file "/usr/local/lib/asir/sp"
258                     return 1/A;
(debug)

This example shows an error caused by zero division in the course of program execution of simpalg(), which attempted to simplify an algebraic number expression of which the denominator is 0.

Function simpalg() also can take a polynomial as its argument and simplifies algebraic numbers in its coefficients.

[43] simpalg(1/A0*x+1/(A0+1));
(-#0)*x+(-1/2*#0+1/2)

Thus, you can operate in polynomials which contain algebraic numbers as you do usually in ordinary polynomials, except for proper simplification by simpalg(). You may sometimes feel needs to convert root’s into indeterminates, especially when you are working for norm computation in algorithms for algebraic factorization. Function algptorat() is used for such cases.

[83] A0=newalg(x^2+1);
(#0)
[84] A1=newalg(x^3+A0*x+A0);
(#1)
[85] T=(2*A0+A1*A0+A1^2)*x+(1+A1)/(2+A0);
(#1^2+#0*#1+2*#0)*x+((#1+1)/(#0+2))
[86] S=algptorat(T);
(((t#0+2)*t#1^2+(t#0^2+2*t#0)*t#1+2*t#0^2+4*t#0)*x+t#1+1)/(t#0+2)
[87] algptorat(coef(T,1));
t#1^2+t#0*t#1+2*t#0

As you see by the example, function algptorat() converts root’s, #n, in polynomials and numbers into its associated indeterminates, t#n. As was already mentioned those indeterminates cannot be directly input in their immediate form. The restriction is adopted to avoid the confusion that might happen if the user could input such internally generatable indeterminates.

The associated indeterminate to a root is reversely converted into the root by rattoalgp() function.

[88] rattoalgp(S,[alg(0)]);
(((#0+2)/(#0+2))*t#1^2+((#0^2+2*#0)/(#0+2))*t#1
+((2*#0^2+4*#0)/(#0+2)))*x+((1)/(#0+2))*t#1+((1)/(#0+2))
[89] rattoalgp(S,[alg(0),alg(1)]);
(((#0^3+6*#0^2+12*#0+8)*#1^2+(#0^4+6*#0^3+12*#0^2+8*#0)*#1
+2*#0^4+12*#0^3+24*#0^2+16*#0)/(#0^3+6*#0^2+12*#0+8))*x
+(((#0+2)*#1+#0+2)/(#0^2+4*#0+4))
[90] rattoalgp(S,[alg(1),alg(0)]);
(((#0+2)*#1^2+(#0^2+2*#0)*#1+2*#0^2+4*#0)/(#0+2))*x
+((#1+1)/(#0+2))
[91] simpalg(@89);
(#1^2+#0*#1+2*#0)*x+((-1/5*#0+2/5)*#1-1/5*#0+2/5)
[92] simpalg(@90);
(#1^2+#0*#1+2*#0)*x+((-1/5*#0+2/5)*#1-1/5*#0+2/5)

Function rattoalgp() takes as the second argument a list consisting of root’s that you want to convert, and converts them successively from the left. This example shows that apparent difference of the results due to the order of such conversion will vanish by simplification yielding the same result. Functions algptorat() and rattoalgp() can be conveniently used for your own simplification.