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Dickson's lemma states that, for a fixed $k in mathbf N^+$, every set of $k$-tuples of natural numbers has finitely many elements that are minimal with respect to the product order induced on $mathbf N^k$ from the integers.

The lemma has many and diverse applications: E.g., it was used by L.E. Dickson in his work on odd perfect numbers (that's where the lemma takes its name from); and by P. Gordan as part of his 1899 proof of Hilbert's basis theorem (though it is probably more common to use Hilbert's basis theorem to prove Dickson's lemma, and hence to regard the former as a generalization of the latter). Further applications include finiteness results in factorization theory, where the lemma is used, e.g., to show that every atomic, cancellative, commutative monoid that is finitely generated up to units, has accepted elasticity (with all the nice things this implies for a wide spectrum of arithmetical invariants including local elasticities, distances, unions of sets of lengths, etc.).

Question. What are some striking applications of Dickson's lemma you have encountered in your research? One answer per post, and please
provide some context (if possible).

This post should be community wiki, but I don't know how to make it so.

As Darij said in the comments, Dickson's lemma and its variants often come up in Gröbner theory. One application of this is Sam and Snowden's proof of the Noetherian property for various categories (like $FI$) which arise in representation stability.

As an immediate corollary one recovers the eventual stability of Kronecker coefficients (structure constants for tensor products of symmetric group representations).

In theoretical computer science, Dickson's lemma (or one of its equivalent forms) is frequently used to show that various problems involving the behavior of structured transition systems are decidable. One example is the coverability problem for vector addition systems: given an alphabet of vectors $ v_1, ldots , v_k subset mathbbZ^n$ and starting/target vectors $v_0, uin mathbbZ_ge 0^n$, we're looking for a sequence $v_i_1,ldots , v_i_k$ such that $v_0 + v_1 + ldots + v_i_jge 0$ for all $j$ and $v_0 + v_1 + ldots + v_i_kge u$---or a certificate that no such sequence exists. Rackoff proved that, if such a sequence exists, we may take $k$ to be doubly exponential in $n.$ [1] The proof technique can be understood in terms of ascending chains of monomial ideals---in another language, these are descending chains of order ideals. [2]

The complexity of this and related problems are connected to the following effective variant of Hilbert's basis theorem: how long can a strictly ascending chain of ideals $$I_1subsetneq I_2 subsetneq cdots subsetneq I_l$$ in $k[x_1,ldots , x_n]$ be? To make some sense of the question we should control the degrees of the generators of each $I_i$---assume we have some explicit function $f: mathbbN to mathbbN$ such that each $I_i$ is generated in degree $f(i).$ For each fixed $n,$ if the function $f$ is primitive recursive, then there exists a bound (depending primitive-recursively on $f$) on the length of such a chain. [3] However, the dependence of this bound on $n,$ even when $f(i)$ grows linearly in $i,$ is generally not primitive recursive! [4] This is somewhat remarkable, given that we "expect" doubly-exponential bounds for similar-looking questions.

This problem was also studied in the context of polynomial dynamical systems. [5] In the case of interest to the authors, they are able to determine better bounds (doubly exponential) by exploiting the fact that the dimensions of successive colon ideals $dim (I_k:I_k+1)$ are non-increasing.

Some personal context is that these problems came up in my master's research---part of which was incorporated into the paper [6].

[1] Rackoff, Charles. "The covering and boundedness problems for vector addition systems." Theoretical Computer Science 6.2 (1978): 223-231.

[2] Lazić, Ranko, and Sylvain Schmitz. "The ideal view on Rackoff’s coverability technique." International Workshop on Reachability Problems. Springer, Cham, 2015.

[3] Seidenberg, A. "On the length of a Hilbert ascending chain." Proceedings of the American Mathematical Society 29.3 (1971): 443-450.

[4] Socias, Guillermo Moreno. "Length of polynomial ascending chains and primitive recursiveness." Mathematica Scandinavica (1992): 181-205.

[5] Novikov, Dmitri, and Sergei Yakovenko. "Trajectories of polynomial vector fields and ascending chains of polynomial ideals." Annales de l'institut Fourier. Vol. 49. No. 2. 1999.

[6] Benedikt, Michael, et al. "Polynomial automata: Zeroness and applications." Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science. IEEE Press, 2017.