# Stillman’s conjecture and Gröbner bases

Start with linear forms $f_1,\dots,f_k$ in infinitely many variables $\{x_i\mid i>0\}$. Fix the monomial order to be the graded reverse lexicographic (with $x_1>x_2>\cdots$). Then the reduced Gröbner basis (GB) for the input sequence $f$ is equivalent to the row-reduced echelon form (RREF) of the $k \times \infty$ matrix of the coefficients of $f$.

Suppose one makes a generic linear change of variables prior to computing GB (RREF). Then the dimension $r$ of the subspace spanned by $f$ determines the leading monomials of the GB (pivots in the RREF): $\{x_1,\dots,x_r\}$. Note that there is a finite number of outcomes as $r$  is at most $k$.

In full analogy to the above, stepping into the nonlinear land, [DLL] establishes that there is a finite number of generic initial ideals (gins) for the ideal $\langle f_1,\dots,f_k \rangle$ for the homogeneous generators of fixed arbitrary degrees $d_1,\dots,d_k$ (the linear case is $d_1=\dots=d_k=1$). Moreover, this remains true beyond polynomials: $f$ may come from a big ring of formal power series of bounded degree.

The finiteness of the set of gins implies yet another proof of Stillman’s conjecture:

The projective dimension of an ideal generated by homogeneous forms of fixed degrees but in an arbitrary number of variables and in arbitrary characteristic is uniformly bounded.

Notably, unlike in the linear case, the characteristic of the field of coefficients may make a difference for the outcome (i.e., the set of possible gins). However, the union of these over all characteristics is still finite!

The following articles (listed chronologically) contain proofs of Stillman’s conjecture.

[AH]  Ananyan, Hochster. “Small subalgebras of polynomial rings and Stillman’s conjecture.”

[ESS]  Erman, Snowden, Sam. “Big polynomial rings and Stillman’s conjecture.”

[DLL]  Draisma, Lasoń, Leykin. “Stillman’s conjecture via generic initial ideals.”

In principle, it is possible to use the bounds on the projective dimension given in [AH] for an effective approach to find all gins. One can also implement the algorithm of [DLL] on a computer. While the latter may seem more efficient than the former, neither approach may lead to computational results that surpass those obtained in special cases by humans.

(Disclaimer: the last statement represents the author’s opinion of the capabilities of humans and machines at the moment of writing.)