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# Arithmetic Circuit Complexity of Division and Truncation

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## Acknowledgements

We thank Himanshu Shukla for several discussions on the complexity of truncated power series, and for bringing the reference [Coons and Borwein, 2008] to our attention. P. D. would like to thank CSE, IIT Kanpur for the hospitality. A. S. would like to thank the Institute of Theoretical Computer Science at Ulm University for the hospitality. We thank Thomas Thierauf and Nitin Saxena for discussions and feedback on the draft.

## Cite As

Pranjal Dutta, Gorav Jindal, Anurag Pandey, and Amit Sinhababu. Arithmetic Circuit Complexity of Division and Truncation. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 25:1-25:36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.CCC.2021.25

## Abstract

Given polynomials f,g,h ∈ 𝔽[x₁,…,x_n] such that f = g/h, where both g and h are computable by arithmetic circuits of size s, we show that f can be computed by a circuit of size poly(s,deg(h)). This solves a special case of division elimination for high-degree circuits (Kaltofen'87 & WACT'16). The result is an exponential improvement over Strassen’s classic result (Strassen'73) when deg(h) is poly(s) and deg(f) is exp(s), since the latter gives an upper bound of poly(s, deg(f)). Further, we show that any univariate polynomial family (f_d)_d, defined by the initial segment of the power series expansion of rational function g_d(x)/h_d(x) up to degree d (i.e. f_d = g_d/h_d od x^{d+1}), where circuit size of g is s_d and degree of g_d is at most d, can be computed by a circuit of size poly(s_d,deg(h_d),log d). We also show a hardness result when the degrees of the rational functions are high (i.e. Ω (d)), assuming hardness of the integer factorization problem. Finally, we extend this conditional hardness to simple algebraic functions as well, and show that for every prime p, there is an integral algebraic power series with its minimal polynomial satisfying a degree p polynomial equation, such that its initial segment is hard to compute unless integer factoring is easy, or a multiple of n! is easy to compute. Both, integer factoring and computation of multiple of n!, are believed to be notoriously hard. In contrast, we show examples of transcendental power series whose initial segments are easy to compute.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Algebraic complexity theory
• Theory of computation → Computational complexity and cryptography
##### Keywords
• Arithmetic Circuits
• Division
• Truncation
• Division elimination
• Rational function
• Algebraic power series
• Transcendental power series
• Integer factorization

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