Solving a system of polynomial equations - can I be sure that the number of solutions, and the solutions are correct?

Question

I need to solve a system of two polynomials with integer coefficients in two variables, $\{Q_1(w,z)=0,\,Q_2(w,z)=0\}$, and want to compute all real solutions. I'm able to run Solve in Mathematica and find that there are 69 solutions, 13 of which are real.

My questions are:

1. Can I fully trust that Mathematica correctly finds the number of real solutions to this system, and that these solutions are correct?

2. If so, what theoretical results ensure that the Solve algorithm finds all the solutions to the system?

I read the documentation about solving polynomial systems here and here. Unfortunately I don't have the necessary background in algebra (e.g., Grobner basis) to understand the algorithm.

Here is the code I used, thank you very much for your help.

First we define the polynomials

Q1[w_, z_] := -168 w^11 - 1386 w^10 z + w^9 (730 - 420 z^2) - 
  9 w^8 z (-489 + 370 z^2) + 14 w^6 z (-135 + 256 z^2 + 6 z^4) + 
  6 w^2 z^5 (747 - 282 z^2 + 29 z^4) - 
  8 w^3 z^4 (513 + 209 z^2 + 39 z^4) - 16 w^7 (45 - 26 z^2 + 72 z^4) +
   z^7 (270 + 79 z^2 + 78 z^4) + 
  30 w^4 z^3 (-249 - 38 z^2 + 138 z^4) - 
  2 w z^6 (-1134 + 67 z^2 + 186 z^4) - 
  12 w^5 z^2 (171 - 363 z^2 + 298 z^4)

Q2[w_, z_] := -126 w^11 - 84 w^10 z + w^9 (489 - 1110 z^2) - 
  8 w^8 z (-13 + 72 z^2) + 12 z^7 (216 - 65 z^2 + 4 z^4) + 
  6 w^7 (-45 + 256 z^2 + 10 z^4) - 
  12 w^4 z^3 (342 + 209 z^2 + 52 z^4) + 
  6 w^3 z^4 (1245 - 658 z^2 + 87 z^4) + 
  3 w z^6 (630 + 237 z^2 + 286 z^4) - 
  12 w^6 z (57 - 242 z^2 + 298 z^4) - 
  4 w^2 z^5 (-1701 + 134 z^2 + 465 z^4) + 
  6 w^5 z^2 (-747 - 190 z^2 + 966 z^4)

Then, use Solve to obtain the number of solutions, and then restrict to real solutions

allsols = Solve[{Q1[w, z] == 0, Q2[w, z] == 0}, {z, w}]
Length[allsols]
(* 69 *)

realsols = Solve[{Q1[w, z] == 0, Q2[w, z] == 0}, {z, w}, Reals]
Length[realsols]
(* 13 *)

Finally, I want to check if any solutions satisfies $0<w<1$ and $z>0$. I find that none of the solutions satisfy these conditions.

restrsols = Solve[{Q1[w, z] == 0, Q2[w, z] == 0, 0 < w < 1, z > 0}, {z, w}, Reals]
Length[restrsols]
(* 0 *)

Answer 1

I believe you missed a lot of solutions, namely, infinitely many of them.

As Daniel points out in the comment below, there is a multiple root at (0,0). Apart from that no roots are missed. Have a look

realsols = Solve[{Q1[w, z] == 0, Q2[w, z] == 0}, {z, w}, Reals]
intersections={w, z} /. realsols//N
ContourPlot[{Q1[w,z]==0,Q2[w,z]==0},{w,-2,2},{z,-2,2},
                                    PlotPoints->100,
                                    Epilog -> {Red, PointSize[Large], Point@intersections}]

And now we zoom in

ContourPlot[{Q1[w,z]==0,Q2[w,z]==0},
             {w,0.015,0.02},{z,-0.02,-0.015},
             PlotPoints->100,ContourStyle->{Thickness[0.02],Red}]

The interpretation is likewise can be found in the Daniel comment, namely, "the zero sets intersect tangentially and very slowly separate". But they are not the roots.