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The Maxima on-line user's manual

Algebra Calculator

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Direct Calculator

Direct

Function: direct ([<p_1>, ..., <p_n>], <y>, <f>, [<lvar_1>, ..., <lvar_n>]) calculates the direct image (see M. Giusti, D. Lazard et A. Valibouze, ISSAC 1988, Rome) associated to the function <f>, in the lists of variables <lvar_1>, ..., <lvar_n>, and in the polynomials <p_1>, ..., <p_n> in a variable <y>. The arity of the function <f> is important for the calulation. Thus, if the expression for <f> does not depend on some variable, it is useless to include this variable, and not including it will also considerably reduce the amount of computation.

l: [[x, y], [a, b]];
 pui_direct (multi_orbit (a*x + b*y, l), l, [2, 2]);
 pui_direct (multi_orbit (a*x + b*y, l), l, [3, 2]);
 pui_direct ([y + x + 2*c, y + x + 2*b, y + x + 2*a],                [[x, y], [a, b, c]], [2, 3]);

          (%i1) direct ([z^2  - e1* z + e2, z^2  - f1* z + f2],
                        z, b*v + a*u, [[u, v], [a, b]]);
                 2
          (%o1) y  - e1 f1 y

2 2 2 2 - 4 e2 f2 - (e1 - 2 e2) (f1 - 2 f2) + e1 f1 + ----------------------------------------------- 2

          (%i2) ratsimp (%);
                        2                2                   2
          (%o2)        y  - e1 f1 y + (e1  - 4 e2) f2 + e2 f1
          (%i3) ratsimp (direct ([z^3-e1*z^2+e2*z-e3,z^2  - f1* z + f2],
                        z, b*v + a*u, [[u, v], [a, b]]));
                 6            5         2                        2    2   4
          (%o3) y  - 2 e1 f1 y  + ((2 e1  - 6 e2) f2 + (2 e2 + e1 ) f1 ) y

3 3 3 + ((9 e3 + 5 e1 e2 - 2 e1 ) f1 f2 + (- 2 e3 - 2 e1 e2) f1 ) y

2 2 4 2 + ((9 e2 - 6 e1 e2 + e1 ) f2

2 2 2 2 4 + (- 9 e1 e3 - 6 e2 + 3 e1 e2) f1 f2 + (2 e1 e3 + e2 ) f1 )

2 2 2 3 2 y + (((9 e1 - 27 e2) e3 + 3 e1 e2 - e1 e2) f1 f2

2 2 3 5 + ((15 e2 - 2 e1 ) e3 - e1 e2 ) f1 f2 - 2 e2 e3 f1 ) y

2 3 3 2 2 3 + (- 27 e3 + (18 e1 e2 - 4 e1 ) e3 - 4 e2 + e1 e2 ) f2

2 3 3 2 2 + (27 e3 + (e1 - 9 e1 e2) e3 + e2 ) f1 f2

2 4 2 6 + (e1 e2 e3 - 9 e3 ) f1 f2 + e3 f1

Finding the polynomial whose roots are the sums a+u where a is a root of z^2 - e_1 z + e_2 and u is a root of z^2 - f_1 z + f_2.

          (%i1) ratsimp (direct ([z^2 - e1* z + e2, z^2 - f1* z + f2],
                                    z, a + u, [[u], [a]]));
                 4                    3             2
          (%o1) y  + (- 2 f1 - 2 e1) y  + (2 f2 + f1  + 3 e1 f1 + 2 e2

2 2 2 2 + e1 ) y + ((- 2 f1 - 2 e1) f2 - e1 f1 + (- 2 e2 - e1 ) f1

2 2 2 - 2 e1 e2) y + f2 + (e1 f1 - 2 e2 + e1 ) f2 + e2 f1 + e1 e2 f1

2 + e2

direct accepts two flags: elementaires and puissances (default) which allow decomposing the symmetric polynomials appearing in the calculation into elementary symmetric functions, or power functions, respectively.

Functions of sym used in this function:

multi_orbit (so orbit), pui_direct, multi_elem (so elem), multi_pui (so pui), pui2ele, ele2pui (if the flag direct is in puissances).

There are also some inexact matches for direct. Try ?? direct to see them.

(%o1)                                true
(%i2) 

Direct Example

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