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Counting Real Roots in Polynomial-Time via Diophantine Approximation.

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Foundations of Computational Mathematics (New York, N.Y.)
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PubMed
Summary
This summary is machine-generated.

This study presents the first algorithm for precisely counting real roots in polynomial systems. The method achieves polynomial time complexity with respect to the number of terms for fixed dimensions.

Keywords:
Baker–Wustholtz theoremCircuitDescartes’ ruleGale dualMahler’s theoremPositive rootReal rootRolle’s theoremSparse polynomial system

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Area of Science:

  • Algebraic Geometry
  • Computational Mathematics
  • Computer Science

Background:

  • Counting real roots of polynomial systems is a fundamental problem in computational algebraic geometry.
  • Existing methods often struggle with high-dimensional systems or lack guaranteed polynomial-time complexity.
  • The complexity is often dependent on the degree and number of variables, which can be computationally prohibitive.

Purpose of the Study:

  • To develop a novel algorithm for exactly counting the real roots of polynomial systems.
  • To achieve a time complexity that is polynomial in the number of terms for a fixed dimension.
  • To explore potential further optimizations based on number-theoretic hypotheses.

Main Methods:

  • The study introduces a new counting algorithm for real roots of polynomial systems.
  • The algorithm's runtime is analyzed and shown to be polynomial in the number of terms for fixed dimensions.
  • The analysis considers systems with generic integer coefficients and bounded exponent vectors.

Main Results:

  • The paper provides the first algorithm that counts the exact number of real roots of a polynomial system in time polynomial in the number of terms, for any fixed dimension.
  • The algorithm's efficiency is demonstrated for systems with specific geometric and coefficient constraints.
  • A number-theoretic hypothesis is proposed that could lead to further improvements in computational efficiency.

Conclusions:

  • The developed algorithm offers a significant advancement in the computational study of real roots for polynomial systems.
  • This work lays the groundwork for more efficient root-counting methods in algebraic geometry.
  • Future research may leverage number theory to accelerate these computations further.