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Abstract. We study the distribution of complex zeros of Gaussian harmonic polynomials with independent complex coefficients. The expected number of zeros is evaluated by applying a formula of independent interest for the expected absolute value of quadratic forms of Gaussian random variables.

We provide general formulas to compute the expectations of absolute value and sign of Gaussian quadratic forms, i.e. E | X, AX + b, X + c| and E sgn( X, AX + b, X + c) for centered Gaussian random vector X, fixed matrix A, vector b and constant c. Products of Gaussian quadratics are also discussed and followed with several interesting applications.

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