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Resolvante

Function: resolvante (<P>, <x>, <f>, [<x_1>,..., <x_d>]) 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: * A polynomial in a single variable,

* linear,

* alternating,

* a sum,

* symmetric,

* a product,

* the function of the Cayley resolvent (usable up to degree 5)

(x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x1 - (x1*x3 + x3*x5 + x5*x2 + x2*x4 + x4*x1))^2

general, the flag of resolvante may be, respectively: * unitaire,

* lineaire,

* alternee,

* somme,

* produit,

* cayley,

* generale.

          (%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.

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

(%o1)                                true
(%i2)