Over Brauer fields, systems of degree-d forms with Birch rank above s^{2^{d-1}} C_d have Zariski dense solution sets, with C_d an explicit product of diagonal-form thresholds.
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Density of solutions for systems of forms
Over Brauer fields, systems of degree-d forms with Birch rank above s^{2^{d-1}} C_d have Zariski dense solution sets, with C_d an explicit product of diagonal-form thresholds.