[Paper Review] Some conjectures on addition and multiplication of complex (real) numbers

Apoloniusz Tyszka|arXiv (Cornell University)|Jul 18, 2008

Mathematical Dynamics and Fractals7 citations

TL;DR

This paper investigates conjectures on bounded rational and complex number solutions for systems of linear and polynomial equations derived from addition and multiplication operations. It proposes that solutions exist within exponentially bounded intervals—specifically, $[-2^{n-1}, 2^{n-1}]$ for linear systems and $[0, 2^{2^{n-2}}]$ for systems including multiplication—supported by probabilistic verification and theoretical analysis using matrix algebra and pseudoinverses.

ABSTRACT

We discuss conjectures related to the following two conjectures: (1) for each complex numbers x_1,...,x_n there exist rationals y_1,...,y_n \in [-2^{n-1},2^{n-1}] such that \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in {1,...,n} (x_i+x_j=x_k \Rightarrow y_i+y_j=y_k) (2) for each complex (real) numbers x_1,...,x_n there exist complex (real) numbers y_1,...,y_n such that \forall i \in {1,...,n} |y_i| \leq 2^{2^{n-2}} \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in {1,...,n} (x_i+x_j=x_k \Rightarrow y_i+y_j=y_k) \forall i,j,k \in {1,...,n} (x_i \cdot x_j=x_k \Rightarrow y_i \cdot y_j=y_k)

Motivation & Objective

• To establish tight upper bounds on rational and complex number solutions for systems of equations involving addition and multiplication.
• To investigate whether solutions to consistent systems of linear and polynomial equations can be confined within exponentially bounded intervals.
• To verify the minimality and tightness of the proposed bounds using theoretical analysis and probabilistic computation.
• To explore the role of Moore-Penrose pseudoinverses and least-squares solutions in deriving bounded rational solutions.
• To examine the implications of these bounds for maximal and finite-solution systems in complex and real number domains.

Proposed method

• Formalizes systems $S \subseteq W_n$ consisting of equations $x_i = 1$ and $x_i + x_j = x_k$, and extends them to include $x_i \cdot x_j = x_k$ for polynomial systems.
• Applies Cramer’s rule and matrix rank analysis to show that unique solutions to consistent systems are rational when entries are in \{-1, 0, 1, 2\}.
• Uses Moore-Penrose pseudoinverse $\mathbf{A}^\dagger$ to compute the least-squares solution with minimal Euclidean norm, ensuring rational entries.
• Employs probabilistic algorithms in MuPAD to sample random systems and verify that solution norms remain within $[-2^{n-1}, 2^{n-1}]$ and $[0, 2^{2^{n-2}}]$.
• Reduces systems by substituting variables (e.g., replacing $x_i = 1$ with $x_1$) to simplify proof and verify conjectures for smaller $n$.
• Analyzes maximal and finite-solution systems to test conjectures on solution bounds, using Gröbner basis dimension checks and numeric solving.

Experimental results

Research questions

• RQ1 Can every consistent system of linear equations over $\mathbb{C}$ with $x_i = 1$ and $x_i + x_j = x_k$ have a rational solution within $[-2^{n-1}, 2^{n-1}]$?
• RQ2 Is the bound $2^{2^{n-2}}$ tight for complex or real solutions of systems including both addition and multiplication equations?
• RQ3 Do least-squares solutions with minimal Euclidean norm for such systems always lie within the proposed exponential bounds?
• RQ4 For systems with finitely many solutions, do all solutions satisfy $|x_i| \leq 2^{2^{n-2}}$?
• RQ5 Can the bound $2^{2^{n-1}}$ be improved for systems with only multiplication and addition, or is it optimal?

Key findings

• Conjecture (I) holds for $n \leq 4$, with a probabilistic confirmation via MuPAD for $n=5$ using 1000 iterations.
• The system $x_1 = 1$, $x_i + x_i = x_{i+1}$ has a unique solution $(1, 2, 4, \dots, 2^{n-1})$, proving that $2^{n-1}$ is the tightest possible bound in Conjecture (I).
• For $n > 1$, the bound $2^{2^{n-2}}$ in Conjecture (II) is tight, as shown by the system $x_1 = 2$, $x_i \cdot x_i = x_{i+1}$, which yields the solution $(2, 4, 16, 256, \dots, 2^{2^{n-1}})$.
• The least-squares solution $\mathbf{x}_0 = \mathbf{A}^\dagger \mathbf{b}$ is rational and lies within $[-2^{n-1}, 2^{n-1}]$, supporting Conjecture 1.
• Probabilistic verification confirms that for $n=5$, the maximum norm of solutions across 1000 random systems remains below $2^{n-1}$.
• Conjecture 5d is supported by code modifications showing that finite-solution systems satisfy $|x_i| \leq 2^{2^{n-1}}$, with the bound being tight for $n > 1$.