This is a brief introduction to the 2023 Polymath Jr project that will be run by Alexandra Seceleanu. It concerns polynomials in several variables. We will consider two types of problems.

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Strength of polynomials: A polynomial f is called reducible is f=g·h where g and h are non-constant polynomials and irreducible otherwise. We will think of reducible polynomials as “weak” and irreducible ones as “strong”. To further quantify how strong an irreducible polynomial is, we consider how many terms it requires to write it as a sum of reducible polynomials. More formally, the strength of a polynomial f is the smallest integer r such that f =g₁·h₁ +...+gᵣ·hᵣ with g₁, h₁,...,gᵣ,hᵣ non constant polynomials. Thus reducible polynomials have strength 1. We will be looking to compute the strength of various classes of polynomials. Some interesting polynomials will be suggested for exploration, but participants will be free to choose their own adventure, that is, polynomial.

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Polynomials obtained by permutations of variables: Consider a homogeneous polynomial f in several variables. We can ask what is the size of the largest set of linearly independent polynomials that can be obtained from f by permuting the variables. For example, f(x,y,z)=x²+yz produces  f(y,x,z)=f(y,z,x)=y²+xz, f(z,x,y)=f(z,y,x)=z²+xy and f(x,y,z)=f(x,z,y)=x²+yz. These three polynomials are linearly independent, so the answer is three. But one can find other quadratic polynomials f(x,y,z) for which the answer is different: 1,2,3,4,5 are all possible answers. Can you think of an example f for each? We will be interested in the same counting problem when one starts with two or more polynomials. We will also be interested in imposing restrictions on the staring polynomials such as being monomials or binomials. At a more abstract level we will be considering various algebraic structures (ideals and rings) built from such collections of polynomials.

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Software

There will be room for both theoretical and computational work in this project. We will use a computer algebra system called Macaulay2. It is not strictly necessary, but it may be helpful for participants in this group to be familiar with linear algebra or abstract algebra. 

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Once you join the project you will get suggestions for further reading, some problems to start with and possible approaches.

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