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The strength of a homogeneous polynomial (or form) is the smallest length ofan additive decomposition expressing it whose summands are reducible forms.Using polynomial functors, we show that the set of forms with bounded strengthis not always Zariski-closed. More specifically, if the ground field isalgebraically closed, we prove that the set of quartics with strength $\leq3$is not Zariski-closed for a large number of variables.

The set of forms with bounded strength is not closed