Generalization of Cantor Pairing Polynomials (Bijective Mapping Among Natural Numbers) from N02 to N0 to Z2 to N0 and N03 to N

TL;DR

This paper extends Cantor pairing polynomials to larger domains and more complex mappings, introducing a polynomial fit for instant bijective mappings among various integer sets, with applications to storage and Diophantine equations.

Contribution

It introduces a new polynomial fitting method for bijective mappings among larger and more complex domains, simplifying the process compared to traditional series-based approaches.

Findings

Extended bijective polynomials for Z2 to N0 and N03 to N0

New polynomial fit method for instant mapping

Applications to storage and Diophantine equations

Abstract

The Cantor pairing polynomials are extended to larger 2D sub-domains and more complex mapping, of which the most important property is the bijectivity. If corners are involved inside (but not the borders of) domain, more than one connected polynomials are necessary. More complex patterns need more complex subsequent application of math series to obtain the mapping polynomials which are more and more inconvenient, although elementary. A tricky polynomial fit is introduced (six coefficients are involved like in the original Cantor polynomials with rigorous but simple restrictions on points chosen) to buy out the regular treatment of math series to find the pairing polynomials instantly. The original bijective Cantor polynomial C1(x,y)= (x2+2xy+y2+3x+y)/2: N02 to N0 (=positive integers) which is 2-fold and runs in zig - zag way along lines x+y=N is extended e.g. to the bijective P(x,y)= 2x2+4sgn(x)sgn(y)xy+2y2-2H(x)sgn(y)x-y+1: Z2 to N0 (with sign and Heaviside functions, Z is integers) running in spiral way along concentric rhombuses, or to the bijective P3D(x,y,z)= [x3+y3+z3 +3(xz2+yz2 +zx2+2xyz +zy2+yx2+xy2) +3(2x2+2y2 +z2+2xz +2yz+4xy) +5x+11y+2z]/6: N03 to N0 which is 6-fold and runs along plains x+y+z=N. Storage device for triangle matrices is also commented as cutting the original Cantor domain to half along with related Diophantine equations.