Testing systems of real quadratic equations for approximate solutions
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Consider systems of equations q_i(x)=0, where q_i: R^n ⟶ R, i=1, …, m, are quadratic forms. Our goal is to tell efficiently systems with many non-trivial solutions or near-solutions x 0 from systems that are far from having a solution. For that, we pick a delta-shaped penalty function F: R⟶ [0, 1] with F(0)=1 and F(y) < 1 for y 0 and compute the expectation of F(q_1(x)) ⋯ F(q_m(x)) for a random x sampled from the standard Gaussian measure in R^n. We choose F(y)=y^-2sin^2 y and show that the expectation can be approximated within relative error 0< ϵ < 1 in quasi-polynomial time (m+n)^O(ln (m+n)-lnϵ), provided each form q_i depends on not more than r real variables, has common variables with at most r-1 other forms and satisfies |q_i(x)| ≤γx^2/r, where γ >0 is an absolute constant. This allows us to distinguish between "easily solvable" and "badly unsolvable" systems in some non-trivial situations.
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