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We consider random systems of equations over the reals, with $m$ equationsand $m$ unknowns $P_i(t)+X_i(t)=0$, $t\in\mathbb{R}^m$, $i=1,...,m$, where the$P_i$'s are non-random polynomials having degrees $d_i$'s (the "signal") andthe $X_i$'s (the "noise") are independent real-valued Gaussian centered randompolynomial fields defined on $\mathbb{R}^m$, with a probability law satisfyingsome invariance properties. For each $i$, $P_i$ and $X_i$ have degree $d_i$.The problem is the behavior of the number of roots for large $m$. We prove thatunder specified conditions on the relation signal over noise, which imply thatin a certain sense this relation is neither too large nor too small, it followsthat the quotient between the expected value of the number of roots of theperturbed system and the expected value corresponding to the centered system(i.e., $P_i$ identically zero for all $i=1,...,m$), tends to zero geometricallyfast as $m$ tends to infinity. In particular, this means that the behavior ofthis expected value is governed by the noise part.

Random systems of polynomial equations. The expected number of roots under smooth analysis