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Let $P:\mathbb{C}^{n}\to \mathbb{C}$ be an irreducible homogenous polynomial.

Is there a geometric or algebra geometric interpretation for the following quantity:

The maximum number $k$ such that there is a $k$ dimensional subvector space $Y$ of $\mathbb{C}^{n}$ which is included in $P=0$?

Moreover, what is this quantity for $Det:M_{n}(\mathbb{C})\simeq \mathbb{C}^{n^{2}}\to \mathbb{C}$?

This question is motivated by the following post:

Finite codimensional subvector space of $C^{*}$ algebras which contains no invertible elements

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    $\begingroup$ I don't understand what you mean by "interpretation". The definition seems geometric enough to me. $\endgroup$
    – Laurent Moret-Bailly
    Feb 10, 2015 at 17:32
  • $\begingroup$ @LaurentMoret-Bailly Is there any known name for this quantity and what are some properties for this? By geometric I mean: it seems that if this quantity would be large, the variete can not be so "cuvrved". So assuming the real version of this question, the first part of my question is about a possible relation between this quantity and for example some upper bound for the absolute value of the sectional curvature at regular points. Or may be some other interpretation... Now is it clear what I mean? $\endgroup$
    – Ali Taghavi
    Feb 10, 2015 at 17:53
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    $\begingroup$ Perhaps your question is related to this question: mathoverflow.net/questions/15106/… $\endgroup$
    – Mahdi Majidi-Zolbanin
    Feb 10, 2015 at 17:55
  • $\begingroup$ @MahdiMajidi-Zolbanin Thank you very much for the interesting link. $\endgroup$
    – Ali Taghavi
    Feb 10, 2015 at 17:59

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