Sets of range uniqueness for multivariate polynomials and linear functions with rank k

Area of Science:

• Mathematics
• Algebra
• Function Theory

Background:

• A set of functions is termed a set of range uniqueness (SRU) if it uniquely identifies functions within a larger set based on their shared domain and range.
• Understanding the minimum size of such sets is critical for theoretical mathematics and practical applications in function approximation.

Purpose of the Study:

• To determine the exact cardinality of the smallest sets of range uniqueness (SRU) for specific function spaces.
• To establish lower and upper bounds for the size of SRUs for real polynomials and linear functions of a given rank.

Main Methods:

• The study involves theoretical analysis of function spaces, specifically real polynomials and linear functions.
• It utilizes set theory and functional analysis principles to derive cardinality bounds for SRUs.
• Mathematical proofs are employed to demonstrate the existence and non-existence of SRUs of certain sizes.

Main Results:

• The research establishes that the minimum cardinality for an SRU for the set of real polynomials of degree at most k in n variables is (k+1)^n.
• It proves that no SRU exists for this polynomial set with fewer than (k+1)^n functions.
• For the set of linear functions of rank k, the study shows the existence of SRUs with a specific cardinality, while demonstrating that no smaller SRUs exist.

Conclusions:

• The precise minimum size for sets of range uniqueness for polynomials and linear functions has been determined.
• These findings provide exact bounds for function identification problems within these mathematical domains.
• The results contribute to the theoretical understanding of function spaces and uniqueness properties.