Finding solutions of polynomials system

Question

Let $f_1,...,f_n$ be a set of polynomials in $x_1,...,x_n$ with rational coefficients. I need to check whether a system $$f_1=a_1,...,f_n=a_n$$ has a real solution for large enough count of points.

To specify, there is 4 polynomials of degree 2 in 4 variables, and 10^4-10^7 values of $a_1,...,a_n$. I tried

Size = 30 (* really 100-300 *)
f[x1_, x2_, y1_, y2_] := x1 + y1 - 1 (* in fact I need polynomial of degree 2, *)
g[x1_, x2_, y1_, y2_] := x2 + y2     (* but it does not work even for degree 1 *)
Equation[a_, b_] := Reduce[f[x1, x2, y1, y2] == 0 && g[x1, x2, y1, y2] == 0 && 
    x1^2 + x2^2 == a && y1^2 + y2^2 == b, {x1, x2, y1, y2}, Reals]
CheckPoint[{a_, b_}] := ! (FindInstance[Equation[a, b], {x1, x2, y1, y2}, Reals] == {})
Points = Select[Tuples[Range[-Size, Size], 2], CheckPoint]
Graphics[Point[#] & /@ Points, Frame -> True, 
    PlotRange -> {{-Size, Size}, {-Size, Size}}]

but it reduces system each time. Is there efficient enough way to do it almost in no time?

Answer 1

Solutions in this example will all have x1 = (1+a-b)/2. You can use that to solve for x2:

Solve[((1 + a - b)/2)^2 + x2^2 == a^2, x2]

(* Out[46]= {{x2 -> -(1/2) Sqrt[-1 - 2 a + 3 a^2 + 2 b + 2 a b - 
     b^2]}, {x2 -> 1/2 Sqrt[-1 - 2 a + 3 a^2 + 2 b + 2 a b - b^2]}} *)

So the condition that there be real-valued solutions is that 1 - 2 a + 3 a^2 + 2 b + 2 a b - b^2>=0. You can check that instead of invoking FindInstance on the full system and it will be much faster.