# The Building Blocks of Polynomial Rings

We discuss several theorems giving bounds for behavior in polynomial rings over a field. A number of the bounds we discuss are consequences of the following result of Tigran Ananyan and the speaker. Given positive integers n, d there exists an integer B(n,d) with the following property: given at most n polynomials of degree at most d in the polynomial ring R in N variables over an algebraically closed field K, they are contained in a polynomial K-subalgebra A of R generated by at most B(n,d) elements such that these elements form a prime sequence in R. Note that B(n,d) does not depend on K or N. It follows that every prime or primary ideal of A has the same property when extended to R. This leads to a proof of a conjecture of Stillman. But it leads to many other results: for example, Caviglia, Chardin, McCullough, Peeva, and Varbaro have used it to prove that if P is a homogeneous prime ideal contained in the square of the homogeneous maximal ideal of a polynomial ring R over an algebraically closed field, the Castlenuovo-Mumford regularity of R/P is bounded by a function of the multiplicity of R/P: this function does not depend on the number of variables nor on the height of P.