This is Elimination.info, produced by Macaulay2, version 1.21

Elimination -- eliminating specified variables, and Sylvester resultant
***********************************************************************

Description
===========

This package contains functions to eliminate variables from an ideal (that is,
intersect an ideal with a subring generated by certain of the variables), and
to compute Sylvester resultants.

It would be nice to implement multivariate resultants, Bezoutians, and sparse
resultants.  Laurent Buse has written code for this in the past. We hope to
provide a package, Resultants, which will do this.  If you are willing to help
write this, please contact the author if this package!

Author
======

  * Michael E. Stillman <mike@math.cornell.edu>

Version
=======

This documentation describes version 1.0 of Elimination.

Source code
===========

The source code from which this documentation is derived is in the file
Elimination.m2 (see common/share/Macaulay2/Elimination.m2 ).

Exports
=======

  * Functions and commands
      * "discriminant" -- see discriminant(RingElement,RingElement),
      * eliminate
      * "resultant" -- see
        resultant(RingElement,RingElement,RingElement),
      * "sylvesterMatrix" -- see
        sylvesterMatrix(RingElement,RingElement,RingElement),
  * Methods
      * discriminant(RingElement,RingElement)
      * "eliminate(List,Ideal)" -- see eliminate,
      * "eliminate(RingElement,Ideal)" -- see eliminate,
      * resultant(RingElement,RingElement,RingElement)
      * sylvesterMatrix(RingElement,RingElement,RingElement)

For the programmer
==================

The object Elimination, is a package (Macaulay2Doc).

discriminant(RingElement,RingElement)
*************************************

Synopsis
========

  * Function: discriminant,
  * Usage:
        discriminant(f,x)
  * Inputs:
      * f, a ring element (Macaulay2Doc), a polynomial
      * x, a ring element (Macaulay2Doc), a variable in the
        same ring
  * Outputs:
      * a ring element (Macaulay2Doc), the discriminant of f
        with respect to x

Description
===========

+---------------------------------------------------------+
|i1 : R = ZZ/101[x,a,b,c,d]                               |
|                                                         |
|o1 = R                                                   |
|                                                         |
|o1 : PolynomialRing                                      |
+---------------------------------------------------------+
|i2 : f = x^2+a*x+b                                       |
|                                                         |
|      2                                                  |
|o2 = x  + x*a + b                                        |
|                                                         |
|o2 : R                                                   |
+---------------------------------------------------------+
|i3 : g = x^2+c*x+d                                       |
|                                                         |
|      2                                                  |
|o3 = x  + x*c + d                                        |
|                                                         |
|o3 : R                                                   |
+---------------------------------------------------------+
|i4 : time eliminate(x,ideal(f,g))                        |
|     -- used 0.00819088 seconds                          |
|                                                         |
|                      2    2             2           2   |
|o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )  |
|                                                         |
|o4 : Ideal of R                                          |
+---------------------------------------------------------+
|i5 : time ideal resultant(f,g,x)                         |
|     -- used 0.00433109 seconds                          |
|                                                         |
|                        2    2             2           2 |
|o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )|
|                                                         |
|o5 : Ideal of R                                          |
+---------------------------------------------------------+
|i6 : sylvesterMatrix(f,g,x)                              |
|                                                         |
|o6 = {-3} | 1 a b 0 |                                    |
|     {-2} | 0 1 a b |                                    |
|     {-3} | 1 c d 0 |                                    |
|     {-2} | 0 1 c d |                                    |
|                                                         |
|             4       4                                   |
|o6 : Matrix R  <--- R                                    |
+---------------------------------------------------------+
|i7 : discriminant(f,x)                                   |
|                                                         |
|        2                                                |
|o7 = - a  + 4b                                           |
|                                                         |
|o7 : R                                                   |
+---------------------------------------------------------+

See also
========

  * resultant
  * eliminate
  * Elimination -- eliminating specified variables, and Sylvester
    resultant

Ways to use this method:
========================

  * discriminant(RingElement,RingElement)

eliminate
*********

Synopsis
========

  * Usage:
        eliminate(v,J)
  * Inputs:
      * v, a ring element (Macaulay2Doc), or a list (Macaulay2Doc),
        a variable or list of variables of a polynomial
        ring R
      * J, an ideal (Macaulay2Doc), in the ring R
  * Outputs:
      * an ideal (Macaulay2Doc), generated by the elements of J
        not involving the variables v

Description
===========

If the ideal J is homogeneous, then an effort is made to use the Hilbert
function to speed up the computation.
+---------------------------------------------------------+
|i1 : R = ZZ/101[x,a,b,c,d]                               |
|                                                         |
|o1 = R                                                   |
|                                                         |
|o1 : PolynomialRing                                      |
+---------------------------------------------------------+
|i2 : f = x^2+a*x+b                                       |
|                                                         |
|      2                                                  |
|o2 = x  + x*a + b                                        |
|                                                         |
|o2 : R                                                   |
+---------------------------------------------------------+
|i3 : g = x^2+c*x+d                                       |
|                                                         |
|      2                                                  |
|o3 = x  + x*c + d                                        |
|                                                         |
|o3 : R                                                   |
+---------------------------------------------------------+
|i4 : time eliminate(x,ideal(f,g))                        |
|     -- used 0.00816266 seconds                          |
|                                                         |
|                      2    2             2           2   |
|o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )  |
|                                                         |
|o4 : Ideal of R                                          |
+---------------------------------------------------------+
|i5 : time ideal resultant(f,g,x)                         |
|     -- used 0.00429073 seconds                          |
|                                                         |
|                        2    2             2           2 |
|o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )|
|                                                         |
|o5 : Ideal of R                                          |
+---------------------------------------------------------+
|i6 : sylvesterMatrix(f,g,x)                              |
|                                                         |
|o6 = {-3} | 1 a b 0 |                                    |
|     {-2} | 0 1 a b |                                    |
|     {-3} | 1 c d 0 |                                    |
|     {-2} | 0 1 c d |                                    |
|                                                         |
|             4       4                                   |
|o6 : Matrix R  <--- R                                    |
+---------------------------------------------------------+
|i7 : discriminant(f,x)                                   |
|                                                         |
|        2                                                |
|o7 = - a  + 4b                                           |
|                                                         |
|o7 : R                                                   |
+---------------------------------------------------------+

One may also switch the order of arguments: the ideal being the first argument.
This usage has been deprecated, and should no longer be used.

Caveat
======

The ring R should not be a quotient ring, or a non-commutative ring.
Additionally, it would be nice to be able to use a DegreeLimit, or to be able
to interrupt the computation.

See also
========

  * resultant
  * discriminant
  * Elimination -- eliminating specified variables, and Sylvester
    resultant

Ways to use eliminate :
=======================

  * "eliminate(List,Ideal)"
  * "eliminate(RingElement,Ideal)"

For the programmer
==================

The object eliminate, is a method function (Macaulay2Doc).

resultant(RingElement,RingElement,RingElement)
**********************************************

Synopsis
========

  * Function: resultant,
  * Usage:
        resultant(f,g,x)
  * Inputs:
      * f, a ring element (Macaulay2Doc),
      * g, a ring element (Macaulay2Doc), in the same
        polynomial ring R as f
      * x, a ring element (Macaulay2Doc), a variable in R
  * Outputs:
      * a ring element (Macaulay2Doc), the Sylvester
        resultant of f and g with respect to the variable x

Description
===========

The elements f and g should be polynomials in the same ring, and x should be a
variable in that ring.  The result is the determinant of the Sylvester matrix,
sylvesterMatrix(f,g,x).  The resultant of f and its derivative with respect to
x is the discriminant, discriminant(f,x).

+-----------------------------------------------------------------------------+
|i1 : R = ZZ[x,a,b,c,d]                                                       |
|                                                                             |
|o1 = R                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : f = x^7+3*x^4+a*x+b                                                     |
|                                                                             |
|      7     4                                                                |
|o2 = x  + 3x  + x*a + b                                                      |
|                                                                             |
|o2 : R                                                                       |
+-----------------------------------------------------------------------------+
|i3 : g = x^8+x^5+c*x+d                                                       |
|                                                                             |
|      8    5                                                                 |
|o3 = x  + x  + x*c + d                                                       |
|                                                                             |
|o3 : R                                                                       |
+-----------------------------------------------------------------------------+
|i4 : time eliminate(ideal(f,g),x)                                            |
|     -- used 5.86365 seconds                                                 |
|                                                                             |
|            7       8     3 5    8     6         3 4      7       3 3 2      |
|o4 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -    |
|     ------------------------------------------------------------------------|
|        6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  |
|     21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +|
|     ------------------------------------------------------------------------|
|       7       4 3        6       4 2           5         4   2         4 2  |
|     6a d + 21a b d - 8a*b d - 85a b c*d + 41a*b c*d + 79a b*c d - 85a*b c d |
|     ------------------------------------------------------------------------|
|          4 3         3 3         2 4           5       6       5   2        |
|     - 15a c d + 90a*b c d - 50a*b c d + 13a*b*c d - a*c d + 37a b*d  -      |
|     ------------------------------------------------------------------------|
|        2 4 2      5   2      2 3   2      2 2 2 2      2   3 2     2 4 2    |
|     20a b d  - 33a c*d  + 66a b c*d  - 78a b c d  + 38a b*c d  - 6a c d  -  |
|     ------------------------------------------------------------------------|
|        3 2 3      3     3     3 2 3     4 4     2 5      5         2 4      |
|     16a b d  + 25a b*c*d  - 9a c d  - 2a d  - 6a b  + 12a b*c + 84a b c -   |
|     ------------------------------------------------------------------------|
|         2 3 2       2 2 3       2   4      6        3 3       6             |
|     328a b c  + 428a b c  - 178a b*c  - 12a d - 102a b d - 10b d +          |
|     ------------------------------------------------------------------------|
|         3 2         5          3   2       4 2        3 3       3 3         |
|     578a b c*d + 22b c*d - 710a b*c d + 12b c d + 178a c d - 68b c d +      |
|     ------------------------------------------------------------------------|
|        2 4         5        4   2       4 2       4   2        3   2        |
|     62b c d - 18b*c d - 250a b*d  - 2a*b d  + 282a c*d  - 40a*b c*d  +      |
|     ------------------------------------------------------------------------|
|           2 2 2          3 2        4 2     2 2 3      2     3      2 2 3   |
|     104a*b c d  - 80a*b*c d  + 18a*c d  + 4a b d  - 78a b*c*d  + 18a c d  + |
|     ------------------------------------------------------------------------|
|        3 4      3 4      2   4        2 4     3 4          5          5    7|
|     87a d  - 17b d  + 37b c*d  - 23b*c d  + 3c d  - 35a*b*d  + 21a*c*d  - d |
|     ------------------------------------------------------------------------|
|            5     4            4          3 2          2 3           4     5 |
|     + 12a*b  - 8a b*c - 152a*b c + 656a*b c  - 1128a*b c  + 612a*b*c  + 8a d|
|     ------------------------------------------------------------------------|
|           2 3         2 2           2   2        2 3        3   2      4 2  |
|     + 188a b d - 1188a b c*d + 1884a b*c d - 612a c d + 532a b*d  - 88b d  -|
|     ------------------------------------------------------------------------|
|         3   2      3   2      2 2 2         2 3             3       2 4     |
|     756a c*d  + 64b c*d  + 24b c d  + 196a*b d  + 156a*b*c*d  - 714a d  +   |
|     ------------------------------------------------------------------------|
|           5        5     5      4        3 2       2 3         4         3  |
|     114b*d  - 54c*d  - 8b  + 96b c - 432b c  + 864b c  - 648b*c  - 120a*b d |
|     ------------------------------------------------------------------------|
|             2               2          3        2   2       2   2       2 3 |
|     + 792a*b c*d - 1512a*b*c d + 648a*c d - 360a b*d  + 648a c*d  - 504b d  |
|     ------------------------------------------------------------------------|
|               3          4        4                                         |
|     - 216b*c*d  + 2052a*d  - 1944d )                                        |
|                                                                             |
|o4 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i5 : time ideal resultant(f,g,x)                                             |
|     -- used 0.0607104 seconds                                               |
|                                                                             |
|              7       8     3 5    8     6         3 4      7       3 3 2    |
|o5 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +  |
|     ------------------------------------------------------------------------|
|        6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  |
|     21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -|
|     ------------------------------------------------------------------------|
|       7       4 3        6       4 2           5         4   2         4 2  |
|     6a d - 21a b d + 8a*b d + 85a b c*d - 41a*b c*d - 79a b*c d + 85a*b c d |
|     ------------------------------------------------------------------------|
|          4 3         3 3         2 4           5       6       5   2        |
|     + 15a c d - 90a*b c d + 50a*b c d - 13a*b*c d + a*c d - 37a b*d  +      |
|     ------------------------------------------------------------------------|
|        2 4 2      5   2      2 3   2      2 2 2 2      2   3 2     2 4 2    |
|     20a b d  + 33a c*d  - 66a b c*d  + 78a b c d  - 38a b*c d  + 6a c d  +  |
|     ------------------------------------------------------------------------|
|        3 2 3      3     3     3 2 3     4 4     2 5      5         2 4      |
|     16a b d  - 25a b*c*d  + 9a c d  + 2a d  + 6a b  - 12a b*c - 84a b c +   |
|     ------------------------------------------------------------------------|
|         2 3 2       2 2 3       2   4      6        3 3       6             |
|     328a b c  - 428a b c  + 178a b*c  + 12a d + 102a b d + 10b d -          |
|     ------------------------------------------------------------------------|
|         3 2         5          3   2       4 2        3 3       3 3         |
|     578a b c*d - 22b c*d + 710a b*c d - 12b c d - 178a c d + 68b c d -      |
|     ------------------------------------------------------------------------|
|        2 4         5        4   2       4 2       4   2        3   2        |
|     62b c d + 18b*c d + 250a b*d  + 2a*b d  - 282a c*d  + 40a*b c*d  -      |
|     ------------------------------------------------------------------------|
|           2 2 2          3 2        4 2     2 2 3      2     3      2 2 3   |
|     104a*b c d  + 80a*b*c d  - 18a*c d  - 4a b d  + 78a b*c*d  - 18a c d  - |
|     ------------------------------------------------------------------------|
|        3 4      3 4      2   4        2 4     3 4          5          5    7|
|     87a d  + 17b d  - 37b c*d  + 23b*c d  - 3c d  + 35a*b*d  - 21a*c*d  + d |
|     ------------------------------------------------------------------------|
|            5     4            4          3 2          2 3           4     5 |
|     - 12a*b  + 8a b*c + 152a*b c - 656a*b c  + 1128a*b c  - 612a*b*c  - 8a d|
|     ------------------------------------------------------------------------|
|           2 3         2 2           2   2        2 3        3   2      4 2  |
|     - 188a b d + 1188a b c*d - 1884a b*c d + 612a c d - 532a b*d  + 88b d  +|
|     ------------------------------------------------------------------------|
|         3   2      3   2      2 2 2         2 3             3       2 4     |
|     756a c*d  - 64b c*d  - 24b c d  - 196a*b d  - 156a*b*c*d  + 714a d  -   |
|     ------------------------------------------------------------------------|
|           5        5     5      4        3 2       2 3         4         3  |
|     114b*d  + 54c*d  + 8b  - 96b c + 432b c  - 864b c  + 648b*c  + 120a*b d |
|     ------------------------------------------------------------------------|
|             2               2          3        2   2       2   2       2 3 |
|     - 792a*b c*d + 1512a*b*c d - 648a*c d + 360a b*d  - 648a c*d  + 504b d  |
|     ------------------------------------------------------------------------|
|               3          4        4                                         |
|     + 216b*c*d  - 2052a*d  + 1944d )                                        |
|                                                                             |
|o5 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i6 : sylvesterMatrix(f,g,x)                                                  |
|                                                                             |
|o6 = {-14} | 1 0 0 3 0 0 a b 0 0 0 0 0 0 0 |                                 |
|     {-13} | 0 1 0 0 3 0 0 a b 0 0 0 0 0 0 |                                 |
|     {-12} | 0 0 1 0 0 3 0 0 a b 0 0 0 0 0 |                                 |
|     {-11} | 0 0 0 1 0 0 3 0 0 a b 0 0 0 0 |                                 |
|     {-10} | 0 0 0 0 1 0 0 3 0 0 a b 0 0 0 |                                 |
|     {-9}  | 0 0 0 0 0 1 0 0 3 0 0 a b 0 0 |                                 |
|     {-8}  | 0 0 0 0 0 0 1 0 0 3 0 0 a b 0 |                                 |
|     {-7}  | 0 0 0 0 0 0 0 1 0 0 3 0 0 a b |                                 |
|     {-14} | 1 0 0 1 0 0 0 c d 0 0 0 0 0 0 |                                 |
|     {-13} | 0 1 0 0 1 0 0 0 c d 0 0 0 0 0 |                                 |
|     {-12} | 0 0 1 0 0 1 0 0 0 c d 0 0 0 0 |                                 |
|     {-11} | 0 0 0 1 0 0 1 0 0 0 c d 0 0 0 |                                 |
|     {-10} | 0 0 0 0 1 0 0 1 0 0 0 c d 0 0 |                                 |
|     {-9}  | 0 0 0 0 0 1 0 0 1 0 0 0 c d 0 |                                 |
|     {-8}  | 0 0 0 0 0 0 1 0 0 1 0 0 0 c d |                                 |
|                                                                             |
|             15       15                                                     |
|o6 : Matrix R   <--- R                                                       |
+-----------------------------------------------------------------------------+
|i7 : discriminant(f,x)                                                       |
|                                                                             |
|           7          6           3 3          6          5           2 3    |
|o7 = 46656a  - 314928a  - 1555848a b  + 823543b  + 708588a  + 9501786a b  -  |
|     ------------------------------------------------------------------------|
|            4              3            3                                    |
|     531441a  - 20575296a*b  + 15116544b                                     |
|                                                                             |
|o7 : R                                                                       |
+-----------------------------------------------------------------------------+

See also
========

  * sylvesterMatrix
  * discriminant
  * eliminate

Ways to use this method:
========================

  * resultant(RingElement,RingElement,RingElement)

sylvesterMatrix(RingElement,RingElement,RingElement)
****************************************************

Synopsis
========

  * Function: sylvesterMatrix,
  * Usage:
        sylvesterMatrix(f,g,x)
  * Inputs:
      * f, a ring element (Macaulay2Doc), a polynomial in a
        ring R
      * g, a ring element (Macaulay2Doc), a polynomial in
        the same ring
      * x, a ring element (Macaulay2Doc), a variable in R
  * Outputs:
      * a matrix (Macaulay2Doc), the Sylvester matrix of f and g
        with respect to x

Description
===========

Its determinant is the resultant of f and g.
+-----------------------------------------------------------------------------+
|i1 : R = ZZ/101[x,a,b,c,d,Degrees=>{1,1,2,1,2}];                             |
+-----------------------------------------------------------------------------+
|i2 : R = ZZ[x,a,b,c,d]                                                       |
|                                                                             |
|o2 = R                                                                       |
|                                                                             |
|o2 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i3 : f = x^7+3*x^4+a*x+b                                                     |
|                                                                             |
|      7     4                                                                |
|o3 = x  + 3x  + x*a + b                                                      |
|                                                                             |
|o3 : R                                                                       |
+-----------------------------------------------------------------------------+
|i4 : g = x^8+x^5+c*x+d                                                       |
|                                                                             |
|      8    5                                                                 |
|o4 = x  + x  + x*c + d                                                       |
|                                                                             |
|o4 : R                                                                       |
+-----------------------------------------------------------------------------+
|i5 : time eliminate(ideal(f,g),x)                                            |
|     -- used 5.96772 seconds                                                 |
|                                                                             |
|            7       8     3 5    8     6         3 4      7       3 3 2      |
|o5 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -    |
|     ------------------------------------------------------------------------|
|        6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  |
|     21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +|
|     ------------------------------------------------------------------------|
|       7       4 3        6       4 2           5         4   2         4 2  |
|     6a d + 21a b d - 8a*b d - 85a b c*d + 41a*b c*d + 79a b*c d - 85a*b c d |
|     ------------------------------------------------------------------------|
|          4 3         3 3         2 4           5       6       5   2        |
|     - 15a c d + 90a*b c d - 50a*b c d + 13a*b*c d - a*c d + 37a b*d  -      |
|     ------------------------------------------------------------------------|
|        2 4 2      5   2      2 3   2      2 2 2 2      2   3 2     2 4 2    |
|     20a b d  - 33a c*d  + 66a b c*d  - 78a b c d  + 38a b*c d  - 6a c d  -  |
|     ------------------------------------------------------------------------|
|        3 2 3      3     3     3 2 3     4 4     2 5      5         2 4      |
|     16a b d  + 25a b*c*d  - 9a c d  - 2a d  - 6a b  + 12a b*c + 84a b c -   |
|     ------------------------------------------------------------------------|
|         2 3 2       2 2 3       2   4      6        3 3       6             |
|     328a b c  + 428a b c  - 178a b*c  - 12a d - 102a b d - 10b d +          |
|     ------------------------------------------------------------------------|
|         3 2         5          3   2       4 2        3 3       3 3         |
|     578a b c*d + 22b c*d - 710a b*c d + 12b c d + 178a c d - 68b c d +      |
|     ------------------------------------------------------------------------|
|        2 4         5        4   2       4 2       4   2        3   2        |
|     62b c d - 18b*c d - 250a b*d  - 2a*b d  + 282a c*d  - 40a*b c*d  +      |
|     ------------------------------------------------------------------------|
|           2 2 2          3 2        4 2     2 2 3      2     3      2 2 3   |
|     104a*b c d  - 80a*b*c d  + 18a*c d  + 4a b d  - 78a b*c*d  + 18a c d  + |
|     ------------------------------------------------------------------------|
|        3 4      3 4      2   4        2 4     3 4          5          5    7|
|     87a d  - 17b d  + 37b c*d  - 23b*c d  + 3c d  - 35a*b*d  + 21a*c*d  - d |
|     ------------------------------------------------------------------------|
|            5     4            4          3 2          2 3           4     5 |
|     + 12a*b  - 8a b*c - 152a*b c + 656a*b c  - 1128a*b c  + 612a*b*c  + 8a d|
|     ------------------------------------------------------------------------|
|           2 3         2 2           2   2        2 3        3   2      4 2  |
|     + 188a b d - 1188a b c*d + 1884a b*c d - 612a c d + 532a b*d  - 88b d  -|
|     ------------------------------------------------------------------------|
|         3   2      3   2      2 2 2         2 3             3       2 4     |
|     756a c*d  + 64b c*d  + 24b c d  + 196a*b d  + 156a*b*c*d  - 714a d  +   |
|     ------------------------------------------------------------------------|
|           5        5     5      4        3 2       2 3         4         3  |
|     114b*d  - 54c*d  - 8b  + 96b c - 432b c  + 864b c  - 648b*c  - 120a*b d |
|     ------------------------------------------------------------------------|
|             2               2          3        2   2       2   2       2 3 |
|     + 792a*b c*d - 1512a*b*c d + 648a*c d - 360a b*d  + 648a c*d  - 504b d  |
|     ------------------------------------------------------------------------|
|               3          4        4                                         |
|     - 216b*c*d  + 2052a*d  - 1944d )                                        |
|                                                                             |
|o5 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i6 : time ideal resultant(f,g,x)                                             |
|     -- used 0.0622791 seconds                                               |
|                                                                             |
|              7       8     3 5    8     6         3 4      7       3 3 2    |
|o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +  |
|     ------------------------------------------------------------------------|
|        6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  |
|     21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -|
|     ------------------------------------------------------------------------|
|       7       4 3        6       4 2           5         4   2         4 2  |
|     6a d - 21a b d + 8a*b d + 85a b c*d - 41a*b c*d - 79a b*c d + 85a*b c d |
|     ------------------------------------------------------------------------|
|          4 3         3 3         2 4           5       6       5   2        |
|     + 15a c d - 90a*b c d + 50a*b c d - 13a*b*c d + a*c d - 37a b*d  +      |
|     ------------------------------------------------------------------------|
|        2 4 2      5   2      2 3   2      2 2 2 2      2   3 2     2 4 2    |
|     20a b d  + 33a c*d  - 66a b c*d  + 78a b c d  - 38a b*c d  + 6a c d  +  |
|     ------------------------------------------------------------------------|
|        3 2 3      3     3     3 2 3     4 4     2 5      5         2 4      |
|     16a b d  - 25a b*c*d  + 9a c d  + 2a d  + 6a b  - 12a b*c - 84a b c +   |
|     ------------------------------------------------------------------------|
|         2 3 2       2 2 3       2   4      6        3 3       6             |
|     328a b c  - 428a b c  + 178a b*c  + 12a d + 102a b d + 10b d -          |
|     ------------------------------------------------------------------------|
|         3 2         5          3   2       4 2        3 3       3 3         |
|     578a b c*d - 22b c*d + 710a b*c d - 12b c d - 178a c d + 68b c d -      |
|     ------------------------------------------------------------------------|
|        2 4         5        4   2       4 2       4   2        3   2        |
|     62b c d + 18b*c d + 250a b*d  + 2a*b d  - 282a c*d  + 40a*b c*d  -      |
|     ------------------------------------------------------------------------|
|           2 2 2          3 2        4 2     2 2 3      2     3      2 2 3   |
|     104a*b c d  + 80a*b*c d  - 18a*c d  - 4a b d  + 78a b*c*d  - 18a c d  - |
|     ------------------------------------------------------------------------|
|        3 4      3 4      2   4        2 4     3 4          5          5    7|
|     87a d  + 17b d  - 37b c*d  + 23b*c d  - 3c d  + 35a*b*d  - 21a*c*d  + d |
|     ------------------------------------------------------------------------|
|            5     4            4          3 2          2 3           4     5 |
|     - 12a*b  + 8a b*c + 152a*b c - 656a*b c  + 1128a*b c  - 612a*b*c  - 8a d|
|     ------------------------------------------------------------------------|
|           2 3         2 2           2   2        2 3        3   2      4 2  |
|     - 188a b d + 1188a b c*d - 1884a b*c d + 612a c d - 532a b*d  + 88b d  +|
|     ------------------------------------------------------------------------|
|         3   2      3   2      2 2 2         2 3             3       2 4     |
|     756a c*d  - 64b c*d  - 24b c d  - 196a*b d  - 156a*b*c*d  + 714a d  -   |
|     ------------------------------------------------------------------------|
|           5        5     5      4        3 2       2 3         4         3  |
|     114b*d  + 54c*d  + 8b  - 96b c + 432b c  - 864b c  + 648b*c  + 120a*b d |
|     ------------------------------------------------------------------------|
|             2               2          3        2   2       2   2       2 3 |
|     - 792a*b c*d + 1512a*b*c d - 648a*c d + 360a b*d  - 648a c*d  + 504b d  |
|     ------------------------------------------------------------------------|
|               3          4        4                                         |
|     + 216b*c*d  - 2052a*d  + 1944d )                                        |
|                                                                             |
|o6 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i7 : sylvesterMatrix(f,g,x)                                                  |
|                                                                             |
|o7 = {-14} | 1 0 0 3 0 0 a b 0 0 0 0 0 0 0 |                                 |
|     {-13} | 0 1 0 0 3 0 0 a b 0 0 0 0 0 0 |                                 |
|     {-12} | 0 0 1 0 0 3 0 0 a b 0 0 0 0 0 |                                 |
|     {-11} | 0 0 0 1 0 0 3 0 0 a b 0 0 0 0 |                                 |
|     {-10} | 0 0 0 0 1 0 0 3 0 0 a b 0 0 0 |                                 |
|     {-9}  | 0 0 0 0 0 1 0 0 3 0 0 a b 0 0 |                                 |
|     {-8}  | 0 0 0 0 0 0 1 0 0 3 0 0 a b 0 |                                 |
|     {-7}  | 0 0 0 0 0 0 0 1 0 0 3 0 0 a b |                                 |
|     {-14} | 1 0 0 1 0 0 0 c d 0 0 0 0 0 0 |                                 |
|     {-13} | 0 1 0 0 1 0 0 0 c d 0 0 0 0 0 |                                 |
|     {-12} | 0 0 1 0 0 1 0 0 0 c d 0 0 0 0 |                                 |
|     {-11} | 0 0 0 1 0 0 1 0 0 0 c d 0 0 0 |                                 |
|     {-10} | 0 0 0 0 1 0 0 1 0 0 0 c d 0 0 |                                 |
|     {-9}  | 0 0 0 0 0 1 0 0 1 0 0 0 c d 0 |                                 |
|     {-8}  | 0 0 0 0 0 0 1 0 0 1 0 0 0 c d |                                 |
|                                                                             |
|             15       15                                                     |
|o7 : Matrix R   <--- R                                                       |
+-----------------------------------------------------------------------------+
|i8 : discriminant(f,x)                                                       |
|                                                                             |
|           7          6           3 3          6          5           2 3    |
|o8 = 46656a  - 314928a  - 1555848a b  + 823543b  + 708588a  + 9501786a b  -  |
|     ------------------------------------------------------------------------|
|            4              3            3                                    |
|     531441a  - 20575296a*b  + 15116544b                                     |
|                                                                             |
|o8 : R                                                                       |
+-----------------------------------------------------------------------------+

See also
========

  * resultant
  * discriminant
  * Elimination -- eliminating specified variables, and Sylvester
    resultant

Ways to use this method:
========================

  * sylvesterMatrix(RingElement,RingElement,RingElement)