Why does the following nonlinear system have 21 solutions? y + x y 2 − x 3 + 2 x z 4 = 0 − x − y 3 − 3 x 2 y + 3 y z 4 = 0 − 5 2 y 2 z 3 − 2 x 2 z 3 − z 7 2 = 0there is one real and twenty complex solutions: ( 0 , 0 , 0 )

Question

Why does the following nonlinear system have 21 solutions?  y  +  x      y    2    −      x    3    +  2  x      z    4    =  0    −  x  −      y    3    −  3      x    2    y  +  3  y      z    4    =  0    −      5    2        y    2        z    3    −  2      x    2        z    3    −            z      7        2    =  0there is one real and twenty complex solutions:   (  0  ,  0  ,  0  )

Gornil2 · Accepted Answer

Mark the solution   (  0  ,  0  ,  0  ) and remove       z    3   from the last equation. The reduced systems for   z  =  0 has two cubic equations, so at most   3  ⋅  3  =  9 solutions.Then z only occurs as       z    4  . Multiply the first equation by   x, the second by   y and express all equations in   a  =      x    2  ,   b  =      y    2  ,   c  =  x  y and   d  =      z    4  , adding the equation   a  b  −      c    2    =  0. Then the system reads as                    c        +                  c          2                −                  a          2                +        2        a        d                            =        0                            −        a        −        b        c        −        3        a        c        +        3        c        d                            =        0                            −        5        b        −        4        a        −        d                            =        0                            a        b        −                  c          2                                    =        0            This system has the Bezout bound   8 for the number of solutions for   (  a  ,  b  ,  c  ,  d  ), each of these will have at most   8 solutions for   (  x  ,  y  ,  z  ). This reduces the estimate to   64  +  9  +  1  =  74 solutions from the   5  ⋅  5  ⋅  7  =  175 or   5  ⋅  5  ⋅  4  +  3  ⋅  3  =  109 for the Bezout bound of the (reduced) original system.

Why does the following nonlinear system have 21 solutions? y + x y 2 </msup>

Answered question

Answer & Explanation

New Questions in Algebra I