Determining a multiset of numbers by linear combinations of its elements

This project proposed our own generalization to the problem of determining a multiset of numbers by the collection of its s-sums, namely, an investigation of the possibility of unambiguous determination of a multiset of numbers by the collection of linear combinations of all its ordered submultisets of a constant size. This work continues many years of research on this topic, so it is relevant. All obtained results are new.

Research goal: to investigate the problem of determining a multiset of numbers by the collection of linear combinations of all its ordered submultisets of constant size.

Throughout the study, we used methods of classical algebra and number theory to prove theorems, as well as a computer experiment that allowed us to propose hypotheses, and to look for examples of ambiguous determination and corresponding particular cases of coefficient vectors in order to assume their general structure and to prove for it the existence of pairs of multisets of numbers generating the same collection of linear combinations.

During the study, we obtained sufficient conditions for unambiguous determination of a multiset of numbers by a collection of linear combinations of ordered two-element submultisets. It allowed us to completely solve this case by investigating this condition. We also constructed examples of collections of numbers and a general structure of the vector of coefficients that generate identical collections of linear combinations of -element ordered submultisets and it was proved.