geometry ag.algebraica: Do more than $ n $ polynomials in $ n $ help variables to extract common roots?

Dice $ n $ independent homogeneous algebraic degree $ 2 $ polynomials in $ mathbb Z[x_1,dots,x_n]$ we can extract at least one common whole root With the theory of exponential elimination. $ SPACE $ Y $ TIME $.

Suppose we have $ m geq n $ independent homogeneous degree $ 2 $ polynomials in $ mathbb Z[x_1,dots,x_n]$ with

  1. any $ n $ of them being algebraically independent

  2. only the common roots of all $ m $ Polynomials are whole roots.

then it helps to improve the extraction of at least one root to subexponential $ SPACE $ Y $ TIME $ complexity where the input size is $ L $ bits (measure of the number of coefficients and number of bits in coefficients) in some $ m = Omega (n ^ {1+ epsilon}) $ for a fixed $ epsilon en (0, infty) $ (Does the Gaussian elimination choose something integral without relevance for nonlinear properties)?