Filip Jonsson Kling: Multiplying in the squarefree algebra

Say that you do not like to look at squares in your polynomials. One way to get rid of them is to work inside the squarefree algebra given by the quotient of the polynomial ring by the squares of the variables. Since this algebra is graded, we can then ask when multiplications by a fixed homogeneous polynomial induces a map that has full rank in the algebra. In this talk, we will prove that if we pick a degree one polynomial at random, then multiplications by any power of it will have full rank in the squarefree algebra using only basic linear algebra. If time permits, we might talk about what happens if you quotient out by higher powers of the variables instead, or what problems arise in positive characteristics.