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I have $n$ real quadratic equations and $n$ real variables, $x_i$, of the following form:

$$\sum_{i\neq j} a_{ijk}x_ix_j+\sum_ib_{ik}x_i+c_k=0 \ \forall k$$

for $i,j,k\in\{1,\dots n\}$; all coefficients are real and I am interested in the existence of a real solution. Is there a condition on the coefficients to guaranty this?

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    $\begingroup$ Some jargon: real Nullstellensatz (due to Krivine, Dubois, and Risler), real ideal, real radical ideal, real Nullstellensatz certificate. $\endgroup$ – Ricardo Buring Jun 17 '19 at 9:07

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