residue SciMax Toolbox resolvante_alternee1

SciMax Toolbox >> resolvante

resolvante

Maxima Function

Calling Sequence

resolvante (P, x, f, [x_1,..., x_d])

Description

calculates the resolvent of the polynomial P in x of degree n >= d by the function f expressed in the variables x_1, ..., x_d. For efficiency of computation it is important to not include in the list [x_1, ..., x_d] variables which do not appear in the transformation function f.

To increase the efficiency of the computation one may set flags in resolvante so as to use appropriate algorithms:

If the function f is unitary:

the flag of resolvante may be, respectively:

(%i1) resolvante: unitaire$
(%i2) resolvante (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x, x^3 - 1,
      [x]);
" resolvante unitaire " [7, 0, 28, 0, 168, 0, 1120, - 154, 7840,
                         - 2772, 56448, - 33880,
413952, - 352352, 3076668, - 3363360, 23114112, - 30494464,
175230832, - 267412992, 1338886528, - 2292126760]
  3       6      3       9      6      3
[x  - 1, x  - 2 x  + 1, x  - 3 x  + 3 x  - 1,
 12      9      6      3       15      12       9       6      3
x   - 4 x  + 6 x  - 4 x  + 1, x   - 5 x   + 10 x  - 10 x  + 5 x
       18      15       12       9       6      3
 - 1, x   - 6 x   + 15 x   - 20 x  + 15 x  - 6 x  + 1,
 21      18       15       12       9       6      3
x   - 7 x   + 21 x   - 35 x   + 35 x  - 21 x  + 7 x  - 1]
[- 7, 1127, - 6139, 431767, - 5472047, 201692519, - 3603982011]
       7      6        5         4          3           2
(%o2) y  + 7 y  - 539 y  - 1841 y  + 51443 y  + 315133 y
                                              + 376999 y + 125253
(%i3) resolvante: lineaire$
(%i4) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]);
" resolvante lineaire "
       24       20         16            12             8
(%o4) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y
                                                    4
                                       + 344489984 y  + 655360000
(%i5) resolvante: general$
(%i6) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]);
" resolvante generale "
       24       20         16            12             8
(%o6) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y
                                                    4
                                       + 344489984 y  + 655360000
(%i7) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3, x4]);
" resolvante generale "
       24       20         16            12             8
(%o7) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y
                                                    4
                                       + 344489984 y  + 655360000
(%i8) direct ([x^4 - 1], x, x1 + 2*x2 + 3*x3, [[x1, x2, x3]]);
       24       20         16            12             8
(%o8) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y
                                                    4
                                       + 344489984 y  + 655360000
(%i9) resolvante :lineaire$
(%i10) resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]);
" resolvante lineaire "
                              4
(%o10)                       y  - 1
(%i11) resolvante: symetrique$
(%i12) resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]);
" resolvante symetrique "
                              4
(%o12)                       y  - 1
(%i13) resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]);
" resolvante symetrique "
                           6      2
(%o13)                    y  - 4 y  - 1
(%i14) resolvante: alternee$
(%i15) resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]);
" resolvante alternee "
            12      8       6        4        2
(%o15)     y   + 8 y  + 26 y  - 112 y  + 216 y  + 229
(%i16) resolvante: produit$
(%i17) resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]);
" resolvante produit "
        35      33         29        28         27        26
(%o17) y   - 7 y   - 1029 y   + 135 y   + 7203 y   - 756 y
         24           23          22            21           20
 + 1323 y   + 352947 y   - 46305 y   - 2463339 y   + 324135 y
          19           18             17              15
 - 30618 y   - 453789 y   - 40246444 y   + 282225202 y
             14              12             11            10
 - 44274492 y   + 155098503 y   + 12252303 y   + 2893401 y
              9            8            7             6
 - 171532242 y  + 6751269 y  + 2657205 y  - 94517766 y
            5             3
 - 3720087 y  + 26040609 y  + 14348907
(%i18) resolvante: symetrique$
(%i19) resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]);
" resolvante symetrique "
        35      33         29        28         27        26
(%o19) y   - 7 y   - 1029 y   + 135 y   + 7203 y   - 756 y
         24           23          22            21           20
 + 1323 y   + 352947 y   - 46305 y   - 2463339 y   + 324135 y
          19           18             17              15
 - 30618 y   - 453789 y   - 40246444 y   + 282225202 y
             14              12             11            10
 - 44274492 y   + 155098503 y   + 12252303 y   + 2893401 y
              9            8            7             6
 - 171532242 y  + 6751269 y  + 2657205 y  - 94517766 y
            5             3
 - 3720087 y  + 26040609 y  + 14348907
(%i20) resolvante: cayley$
(%i21) resolvante (x^5 - 4*x^2 + x + 1, x, a, []);
" resolvante de Cayley "
        6       5         4          3            2
(%o21) x  - 40 x  + 4080 x  - 92928 x  + 3772160 x  + 37880832 x
                                                       + 93392896

For the Cayley resolvent, the 2 last arguments are neutral and the input polynomial must necessarily be of degree 5.

See also:

resolvante_bipartite, resolvante_produit_sym, resolvante_unitaire, resolvante_alternee1, resolvante_klein, resolvante_klein3, resolvante_vierer, resolvante_diedrale.

residue SciMax Toolbox resolvante_alternee1