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On the Expressive Power of Read-Once Determinants

  • N. R. AravindEmail author
  • Pushkar S. Joglekar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9210)

Abstract

We introduce and study the notion of read-k projections of the determinant: a polynomial \(f \in \mathbb {F}[x_1, \ldots , x_n]\) is called a read-k projection of determinant if \(f=det(M)\), where entries of matrix M are either field elements or variables such that each variable appears at most k times in M. A monomial set S is said to be expressible as read-k projection of determinant if there is a read-k projection of determinant f such that the monomial set of f is equal to S. We obtain basic results relating read-k determinantal projections to the well-studied notion of determinantal complexity. We show that for sufficiently large n, the \(n \times n\) permanent polynomial \(Perm_n\) and the elementary symmetric polynomials of degree d on n variables \(S_n^d\) for \(2 \le d \le n-2\) are not expressible as read-once projection of determinant, whereas \(mon(Perm_n)\) and \(mon(S_n^d)\) are expressible as read-once projections of determinant. We also give examples of monomial sets which are not expressible as read-once projections of determinant.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Indian Institute of TechnologyHyderabadIndia
  2. 2.Vishwakarma Institute of TechnologyPuneIndia

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