# Projective Varieties

Takunari Miyazaki (Trinity College, Hartford, Connecticut)

Let $\mathbf{F}$ be a finite field of order $q\gt 0$. An affine $\ell$-space over $\mathbf{F}$, denoted by $\mathbf{A}^{\ell} = \mathbf{F}^{\ell}$, is the set of all $\ell$-tuples of elements of $\mathbf{F}$. For a polynomial $f \in \mathbf{F} [X_1, \ldots, X_{\ell}]$, let $Z(f)$ denote the {\em zeros} of $f$ in $\mathbf{A}^{\ell}$, i.e.,

$Z(f) = \{ (\alpha_1, \ldots, \alpha_{\ell}) \in \mathbf{A}^{\ell} : f(\alpha_1, \ldots, \alpha_{\ell}) = 0 \}.$

The following problem is NP-complete (cf. [Val79, Section 2]). It is a consequence of this fact that MIN-RANK and other related problems in linear and multilinear algebra are NP-complete (cf. [BFS99] and [MW]).

Affine variety ($q$) (AFF-VAR($q$))

• Instance: a polynomial $f \in \mathbf{F}[X_1, \ldots, X_\ell]$.
• Question: $Z(f) \ne \emptyset$?

A projective $\ell$-space over $\mathbf{F}$, denoted by $\mathbf{P}^{\ell} = \mathbf{P}(\mathbf{F})^{\ell}$, is the set of equivalence classes of $(\ell + 1)$-tuples $(\alpha_0, \alpha_1, \ldots, \alpha_\ell)$, where each $\alpha_i \in \mathbf{F}$, not all zeros, under the relation defined by $(\alpha_0, \alpha_1, \ldots, \alpha_\ell) \sim (\lambda\alpha_0, \lambda\alpha_1, \ldots, \lambda\alpha_\ell)$ for all $\lambda \in \mathbf{F}$, $\lambda \ne 0$. An element of $\mathbf{P}^{\ell}$ is often identified by co"ordinate ratios $(\beta_1 : \cdots : \beta_{\ell})$, where each $\beta_i \in \mathbf{F}$. For a homogeneous polynomial $f \in \mathbf{F}[X_1, \ldots, X_{\ell}]$, the {\em zeros} of $f$ in $\mathbf{P}^{\ell}$ is defined by

$Z(f) = \{ (\beta_1 : \cdots : \beta_{\ell}) \in \mathbf{P}^{\ell} : f(\beta_1, \ldots, \beta_{\ell}) = 0 \}.$

Now, consider:

Projective variety ($q$) (PROJ-VAR($q$))

• Instance: a homogenous polynomial $f \in \mathbf{F}[X_1, \ldots, X_{\ell}]$.
• Question: $Z(f) \ne \emptyset$?

To my knowledge, little is known about the complexity of this problem. I have not seen any proof that it is NP-complete. I have been pondering:

Problem: What is the complexity of PROJ-VAR($q$)?

This originally arose while working with J.B. Wilson to investigate the complexity of testing singularity of bilinear maps (cf. [MW]).

• [BFS99] J. F. Buss, G. S. Frandsen and J. O. Shallit, The computational complexity of some problems of linear algebra, J. Comput. System Sci. 58 (1999), 572–596.

• [MW] T. Miyazaki and J. B. Wilson, Linear-size reductions and completeness in algebra, preprint.

• [Val79] L. G. Valiant, Completeness classes in algebra, Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, Atlanta, Apr. 30–May 2, 1979, ACM, New York, 1979, pp. 249–261.

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