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Let $K$ be a field (probably of positive characteristic) and consider the ring $R=K[\![x_1,\dotsc,x_n]\!]$. Suppose we have an ideal $I=(f_1,\dotsc,f_n)$ (with the same $n$ as before). Suppose we also have elements $a_{ij}$ and $k>0$ with $\sum_ja_{ij}f_j=x_i^k$ for all $i$, which proves that $R/I$ has finite dimension over $K$. Standard results from commutative algebra now tell me that the sequence $f_1,\dotsc,f_n$ is regular (and more precisely, that the associated Koszul complex is exact except in degree zero) and that the socle of $R/I$ has dimension one. The standard proofs tend to use prime ideals, injective envelopes and so on, so they are not very constructive. Is there a more direct and equational approach to this?

There is some relevant theory involving Jacobians but it seems not to work well in positive characteristic so I would prefer to avoid it.

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    $\begingroup$ Did you look at Lombardi and Quitté's book on constructive algebra (arXiv version here)? Your question sounds like something they may have done. $\endgroup$
    – Andrej Bauer
    Commented May 22, 2017 at 8:41
  • $\begingroup$ @AndrejBauer I looked at that book before, and did not find anything directly useful. I just looked again more closely and I now think that the discussion of the Bezout matrix on pages 337 and 366 is potentially relevant. However, I think that a substantial amount of additional work would be needed to answer my question. $\endgroup$
    – Neil Strickland
    Commented May 22, 2017 at 9:12
  • $\begingroup$ May be an alternate procedure is as follows. Your condition says $R/I$ is a finite dimensional $k$-vector space. First, you have to find a basis of this constructively. Then an argument used to prove Weierstrass preparation theorem will show that this basis can be lifted (this is not quite constructive, but at least successively improving the basis upto higher and higher oreder) to a basis of $R$ over $S=k[[u_1,\ldots,u_n]], u_i\mapsto f_i$. This gives the exactness of the Koszul. $\endgroup$
    – Mohan
    Commented May 22, 2017 at 20:42
  • $\begingroup$ [A lot of what you write is unfamiliar to me...but in principle there is much constructivity in algebra.] For my Master's thesis I wrote up how one can construct the complex $p$-adic numbers $\mathbb{C}_p$, by first extending the $p$-adic valuation on $\mathbb{Q}$ to the algebraic numbers (result is called $ \mathbb{A}_p$), and then forming $\mathbb{C}_p$ as the metric completion of $ \mathbb{A}_p$. This illustrates a possible strategy: first restrict your question to discrete fields (decidable equality of elements), and then try to lift answers to the completion field. $\endgroup$
    – Franka Waaldijk
    Commented May 23, 2017 at 19:32
  • $\begingroup$ Also, your question calls to mind the wonderful effectivity of Gröbner bases, when dealing with equations and polynomial elimination. But I'm quite unsure if that is what you could mean (can't hurt to mention it I hope). $\endgroup$
    – Franka Waaldijk
    Commented May 23, 2017 at 19:38

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