Arithmetic with multisets, or msets, incorporates negative numbers and integral arithmetic by employing the particle / anti-particle duality that has played such a big role in 20th century physics, pioneered by the great British physicist Paul Dirac.

Once we have integers, we can expand our arithmetic of polynumbers or polynomials in two different ways: one is to introduce integers as possible coefficients, and the other is to introduce them as possible powers, or exponents. This gives us a very different, perhaps even surprising, approach to this larger integral polynomial arithmetical world. We want to think about these objects here not in a calculus view, where a polynomial is considered as a special kind of "function", but rather in a more purely algebraic way involving again ideas from physics in the form of discrete mass distributions or measures.

In this way multiplication corresponds to traditional convolution of measures. And the entire story supports an important new kind of symmetry, that will play a major role in the further development.

NOTE: There's a typo with the last number of the first poly (a 4 instead of a 3) in the first slide: it should be 2+x+3x^4=
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| 2
| 1
| 0
| 0
| 3 not 4 (thanks Federico Rocca!)

Video Contents:
00:00 Introduction
01:33 Integral Polynumbers and Ipoly
04:08 Integers as coefficients
07:37 BIG difference between Integral Polynumbers and functions
14:31 In algebra: avoid the "function " concept!
19:11 IPoly as measures/ densities
25:14 RPoly : Rational Polynumbers
29:40 The reflection symmetry
34:21 Symmetric /alternating integral polynumbers