# 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)?