quadratic coefficient of polynomial

[YangBo17]

The coefficient of polynomial can be quadratic，what is the theoretical basis?

[blegat]

What do you mean exactly by theoretical basis ? It happens often in control that you get quadratic expressions (or bilinear if you have no square). When you are looking for a quadratic polynomial, it gives a Bilinear Matrix Inequality (BMI). When you look for a polynomial of higher degree, you can reformulate it into a BMI using SumOfSquares.jl. The only solver in Julia that can solve BMI is https://jump.dev/JuMP.jl/stable/installation/#Supported-solvers at the moment.
If you use DSOS you would get a QCQP though so it can be solved with Gurobi to global optimality or Ipopt to local optimality.