NDLI: A near-optimal algorithm for computing real roots of sparse polynomials

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A near-optimal algorithm for computing real roots of sparse polynomials

Content Provider	ACM Digital Library
Author	Sagraloff, Michael
Abstract	Let p ∈ Z[x] be an arbitrary polynomial of degree n with k non-zero integer coefficients of absolute value less than $2^{τ}.$ In this paper, we answer the open question whether the real roots of p can be computed with a number of arithmetic operations over the rational numbers that is polynomial in the input size of the sparse representation of p. More precisely, we give a deterministic, complete, and certified algorithm that determines isolating intervals for all real roots of p with $O(k^{3}·log(nτ)·logn)$ many exact arithmetic operations over the rational numbers. When using approximate but certified arithmetic, the bit complexity of our algorithm is bounded by $Õ(k^{4}·n·(τ+k)),$ where Õ(·) indicates the omission of logarithmic factors. Hence, for sufficiently sparse polynomials (i.e. k = $O(log^{c}(nτ))$ for a constant c), the bit complexity is Õ(nτ), which is optimal up to logarithmic factors.
Starting Page	359
Ending Page	366
Page Count	8
File Format	PDF
ISBN	9781450325011
DOI	10.1145/2608628.2608632
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2014-07-23
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Root refinement Bit complexity Sparse polynomials Root isolation Numerical algorithms Arithmetic complexity
Content Type	Text
Resource Type	Article

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3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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