Determinantal Complexities and Field Extensions

  • Youming Qiao
  • Xiaoming Sun
  • Nengkun Yu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8283)

Abstract

Let \(\mathbb{F}\) be a field of characteristic ≠ 2. The determinantal complexity of a polynomial \(P\in\mathbb{F}[x_1, \dots, x_n]\) is defined as the smallest size of a matrix M whose entries are linear polynomials of x i ’s over \(\mathbb{F}\), such that \(P=\det (M)\) as polynomials in \(\mathbb{F}[x_1, \dots, x_n]\). To determine the determinantal complexity of the permanent polynomial is a long-standing open problem.

Let \(\mathbb{K}\) be an extension field of \(\mathbb{F}\); then P can be viewed as a polynomial over \(\mathbb{K}\). We are interested in the comparison between the determinantal complexity of P over \(\mathbb{K}\) (denoted as \(\mathtt{dc}_{\mathbb{K}}(P)\)), and that of P over \(\mathbb{F}\) (denoted as \(\mathtt{dc}_{\mathbb{F}}(P)\)). It is clear that \(\mathtt{dc}_{\mathbb{K}}(P)\leq \mathtt{dc}_\mathbb{F}(P)\), and the question is whether strict inequality can happen. In this note we consider polynomials defined over ℚ. For \(P=x_1^2+\dots+x_n^2\), there exists a constant multiplicative gap between dc (P) and dc (P): we prove dc (P) ≥ n while ⌈n/2⌉ + 1 ≥ dc (P). We also consider additive constant gaps: (1) there exists a quadratic polynomial Q ∈ ℚ[x, y], such that dc (Q) = 3 and \(\mathtt{dc}_{\overline{\mathbb{Q}}}(Q)=2\); (2) there exists a cubic polynomial C ∈ ℚ[x, y] with a rational zero, such that dc (C) = 4 and \(\mathtt{dc}_{\overline{\mathbb{Q}}}(C)=3\). For additive constant gaps, geometric criteria are presented to decide when \(\mathtt{dc}_{\mathbb{Q}}=\mathtt{dc}_{\overline{\mathbb{Q}}}\).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Youming Qiao
    • 1
  • Xiaoming Sun
    • 2
  • Nengkun Yu
    • 3
    • 4
  1. 1.Centre for Quantum TechnologiesThe National University of SingaporeSingapore
  2. 2.Institute of Computing TechnologyChinese Academy of SciencesChina
  3. 3.State Key Laboratory of Intelligent Technology and Systems, Tsinghua National Laboratory for Information Science and Technology, Department of Computer Science and TechnologyTsinghua UniversityBeijingChina
  4. 4.Centre for Quantum Computation and Intelligent Systems (QCIS), Faculty of Engineering and Information TechnologyUniversity of TechnologySydneyAustralia

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