In[]:=
vars={k,x,y,z,u,v,w,p};​​poly1=-2x-4uv+4yp;​​poly2=-9y+3xp;​​poly3=-4z-4*Sqrt[2]uw+4Sqrt[2]wp+4k;​​poly4=-5u+xv+3Sqrt[2]zw;​​poly5=-v+3xu-Sqrt[5]xw;​​poly6=-w+Sqrt[5]xv+Sqrt[2]zu-Sqrt[2]zp;​​poly7=-5p-7xy-3Sqrt[2]zw;​​polys={poly1,poly2,poly3,poly4,poly5,poly6,poly7}
Out[]=
-4uv-2x+4py,3px-9y,4k+4
2
pw-4
2
uw-4z,-5u+vx+3
2
wz,-v+3ux-
5
wx,-w+
5
vx-
2
pz+
2
uz,-5p-7xy-3
2
wz

FindRoot fails to find solutions in an under-determined system

In[]:=
startingPoints=Transpose[{vars,Table[0,Length[vars]]}]
Out[]=
{{k,0},{x,0},{y,0},{z,0},{u,0},{v,0},{w,0},{p,0}}
In[]:=
FindRoot[polys,startingPoints]
FindRoot
:The number of equations does not match the number of variables in FindRoot[polys,startingPoints].
Out[]=
FindRoot[polys,startingPoints]

NSolve finds solutions intersected with an automatically generated hyperplane

In[]:=
NSolve[polys==0,vars,Reals]
NSolve
:Infinite solution set has dimension at least 1. Returning intersection of solutions with -
139306k
118839
-
58857p
39613
+
35780u
39613
+
113347v
118839
-
135062w
118839
+
168848x
118839
-
133381y
118839
-
134851z
118839
== 1.
Out[]=
{{k-2.52671,x0.,y0.,z-0.645497,u-0.603569,v0.,w1.10196,p0.603569},{k-0.738053,x0.,y0.,z-0.645497,u-0.133878,v0.,w0.244426,p0.133878},{k-0.433471,x0.,y0.,z-0.433471,u0.,v0.,w0.,p0.}}

Find solutions by adding an additional constraint

The value of the parameter k is unbounded
In[]:=
Minimize[{k,polys==0},vars]
Minimize
:The minimum is not attained at any point satisfying the given constraints.
Out[]=
{-∞,{kIndeterminate,xIndeterminate,yIndeterminate,zIndeterminate,uIndeterminate,vIndeterminate,wIndeterminate,pIndeterminate}}
In[]:=
Maximize[{k,polys==0},vars]
Maximize
:The maximum is not attained at any point satisfying the given constraints.
Out[]=
{∞,{kIndeterminate,xIndeterminate,yIndeterminate,zIndeterminate,uIndeterminate,vIndeterminate,wIndeterminate,pIndeterminate}}
By setting k to an arbitrary value, the resulting system is likely not under-determined anymore so NDSolve is able to find a finite number of solutions
In[]:=
sols=Table[NSolve[polys==0&&k==n,vars,Reals],{n,-3,3}]
Out[]=
{{{k-3.,x0,y0,z-3.,u0,v0,w0,p0},{k-3.,x0,y0,z-0.645497,u0.675239,v0,w-1.23281,p-0.675239},{k-3.,x0,y0,z-0.645497,u-0.675239,v0,w1.23281,p0.675239}},{{k-2.,x0,y0,z-2.,u0,v0,w0,p0},{k-2.,x0,y0,z-0.645497,u0.512151,v0,w-0.935055,p-0.512151},{k-2.,x0,y0,z-0.645497,u-0.512151,v0,w0.935055,p0.512151}},{{k-1.,x0,y0,z-1.,u0,v0,w0,p0},{k-1.,x0,y0,z-0.645497,u0.26201,v0,w-0.478362,p-0.26201},{k-1.,x0,y0,z-0.645497,u-0.26201,v0,w0.478362,p0.26201}},{{k0,x0,y0,z0,u0,v0,w0,p0}},{{k1.,x0,y0,z1.,u0,v0,w0,p0},{k1.,x0,y0,z0.645497,u0.26201,v0,w0.478362,p-0.26201},{k1.,x0,y0,z0.645497,u-0.26201,v0,w-0.478362,p0.26201}},{{k2.,x0,y0,z2.,u0,v0,w0,p0},{k2.,x0,y0,z0.645497,u0.512151,v0,w0.935055,p-0.512151},{k2.,x0,y0,z0.645497,u-0.512151,v0,w-0.935055,p0.512151}},{{k3.,x0,y0,z3.,u0,v0,w0,p0},{k3.,x0,y0,z0.645497,u-0.675239,v0,w-1.23281,p0.675239},{k3.,x0,y0,z0.645497,u0.675239,v0,w1.23281,p-0.675239},{k3.,x-0.466988,y0.102925,z0.732384,u0.706628,v0.234125,w1.17225,p-0.661205},{k3.,x-0.466988,y-0.102925,z0.732384,u-0.706628,v-0.234125,w-1.17225,p0.661205},{k3.,x0.466988,y0.102925,z0.732384,u-0.706628,v0.234125,w-1.17225,p0.661205},{k3.,x0.466988,y-0.102925,z0.732384,u0.706628,v-0.234125,w1.17225,p-0.661205}}}
We found 23 solutions by varying k ∈ {-3, ..., 3}
In[]:=
Length[Flatten[sols,1]]
Out[]=
23