Noninjectivity of nonzero discriminant polynomials and applications to packing polynomials

Abstract: We show that an integer-valued quadratic polynomial on $\mathbb{R}^2$ can not be injective on the integer lattice points of any subset of $\mathbb{R}^2$ containing an affine convex cone if its discriminant is nonzero. A consequence is the non-existence of quadratic packing polynomials on irrational sectors of $\mathbb{R}^2$. The result also simplifies a classical proof of the Fueter-Pólya Theorem, which states that the two Cantor polynomials are the only quadratic polynomials bijectively mapping $\mathbb{N}_0^2$ onto $\mathbb{N}_0$.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

Registration for the conference is free. Register at cant2021.eventbrite.com.

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