Core Concepts
Exploring the complexity of decision problems related to representing integers as sums of squares using straight-line programs.
Abstract
この記事では、直線プログラムを使用して整数を二乗の和として表現する問題に関連する決定問題の複雑さを探求しました。PosSLPとその変種に焦点を当て、Div2SLPがDegSLPにどのように関連しているかを示しました。さらに、PosSLPの多項式バリアントが無条件でcoNP-hardであることを示しました。
Stats
SLP C of size poly(pn,ℓ) which computes a polynomial PM(W)
Theorem 1.1 (Proposition 1.1 in [ABKPM09])
Theorem 1.2 (Proposition 1.2 in [ABKPM09])
Theorem 1.3 (Theorem 1.2 in [BJ23])
Theorem 2.1 ([Leg97, Gau01, Ank57, Mor58])
Theorem 3.1 ([Dud12, Section 18])
Conjecture 3.1 (Generalized Cram´er conjecture A, [Kou18])
Quotes
"One can construct a SLP C of size poly(pn,ℓ) which computes a polynomial PM(W)"
"An integer n is not 2SoS if and only if the prime-power decomposition of n contains a prime of the form 4k +3 with an odd power."
"Given a straight-line program P of length s computing a polynomial f ∈ Z[x], we compute:"
"The Blum-Shub-Smale (BSS) computational model deals with computations using real numbers."
"Lagrange proved in 1770 that every natural number can be represented as a sum of four non-negative integer squares."