Let $n$ be a positive integer, and $p$ a prime. Any subgroup $H\le \operatorname{GL}\_n(\mathbb{F}\_p)$ acts on the polynomial ring $\mathbb{F}\_p[x\_1,\ldots,x\_n]$ via $A\cdot x\_i=\sum\_j a\_{ji}x\_j$ for all $A=(a\_{ij})\in H$, and $A\cdot f(x\_1,\ldots,x\_n):=f(A\cdot x\_1,\ldots,A\cdot x\_n)$.



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> > How can we compute the $\mathbb{F}\_p$-algebra of invariants $\mathbb{F}\_p[x\_1,\ldots,x\_n]^H$ given generators of $H$?
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For example, take $H=\operatorname{SL}\_n(\mathbb{F}\_p)$. Then if we let


$$L\_{n,s}=\left|\begin{array}{cccc}
x\_1&x\_2&\cdots&x\_n\\
x\_1^{p}&x\_2^{p}&\cdots&x\_n^p\\
\cdots&\cdots&\cdots&\cdots\\
\widehat{x\_1^{p^s}}&\widehat{x\_2^{p^s}}&\cdots&\widehat{x\_n^{p^s}}\\
\cdots&\cdots&\cdots&\cdots\\
x\_1^{p^n}&x\_2^{p^n}&\cdots&x\_n^{p^n}
\end{array}\right|$$


where the $p^s$ row is omitted, the invariants are generated by the Dickson polynomials $\dfrac{L\_{n,s}}{L\_{n,n}}$ for $s=0,\ldots,n-1$.


However, if $H$ is arbitrary (described by a list of generators $h\_1,\ldots,h\_m$), is there some algorithm for computing the invariants? Or perhaps at least an upper bound on the degree of the generators of the invariants so that we can be sure to have found all of them via some computer search?



Invariant polynomials under the action of $H\le\operatorname{GL}_n(\mathbb{F}_p)$
Let me add to Chris' answer that there is indeed a degree bound: $\mathbb{F}[x\_1,\ldots,x\_n]^H$ is generated in degree at most $n (|H| - 1)$. The bound was proved by Peter Symonds in: *On the Castelnuovo-Mumford regularity of rings of polynomial
invariants*, Ann. of Math. (2) 174 (2011), 499–517.


And yes, a general algorithm can be found in Chapter 3 of the book with Derksen. It is implemented in the computer algebra system MAGMA.


