I am trying to generalize an algorithm for the construction of a certain linear combinations of functions $\boldsymbol{f}:x\in\mathbb{R}\mapsto \boldsymbol{f}(x)\in\mathbb{R}^n$ that utilizes Wronskian matrices, to the multidimensional version $\boldsymbol{f}:\boldsymbol{x}\in\mathbb{R}^m\mapsto\boldsymbol{f}(\boldsymbol{x})\in\mathbb{R}^n$ utilizing "some" multidimensional generalization of Wronskian matrices.

Some online research turned up A. I. Petrov, A multidimensional generalization of the Wronskian, Uspekhi Mat. Nauk, 1964, Volume 19, Issue 5, 194– 196, which is written in Russian language.


is there an English (or German) translation of the above linked paper freely available online?

What else can be recommended as freely accessible resources for the definition of multidimensional Wronskian matrices


from the statement "In the case of functions of several variables, there is no one determinant which may properly be taken as the generalization of the wronskian" in M. Green's 1916 paper The Linear Dependence of Functions of Several Variables, and Completely Integrable Systems of Homogeneous Linear Partial Differential Equations it became clear that questions about the multidimensional Wronskian are ill posed.

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    $\begingroup$ I am pretty sure the whole journal was translated into English; the Cold War was a the golden age of Russian language departments in the United States. It was called Russian Math Surveys in English $\endgroup$
    – Ben McKay
    Dec 17, 2022 at 15:31
  • $\begingroup$ @BenMcKay m other hope is that there exists a German translation from the former GDR $\endgroup$ Dec 17, 2022 at 15:49
  • $\begingroup$ @BenMcKay Looking at the website of Russian Mathematical Surveys, I see only three papers in this volume: iopscience.iop.org/issue/0036-0279/19/5 and turpion.org/php/… (I do not know whether the other ones weren't translated at the time - or whether only some of them are available online. Still, I do not have access - our university doesn't seem to have the subscription.) $\endgroup$ Dec 18, 2022 at 18:26
  • $\begingroup$ @MartinSleziak: There is something odd about the paper; if you download from the mathnet.ru link above, the name of the journal doesn't actually appear. Instead it says Mathematical Life in the USSR as the title. So maybe mathnet.ru has not only the author's name wrong, but also the source. I think it was a collection of abstracts celebrating and summarizing recent mathematics in the USSR in 1964. $\endgroup$
    – Ben McKay
    Dec 18, 2022 at 21:56
  • $\begingroup$ I don't know if this is relevant for your desired generalization of the Wronskian, but the following monograph about certain multidimensional generalizations of determinants might be of interest to you: Discriminants, Resultants, and Multidimensional Determinants by Gelfand, Kapranov & Zelevinsky (Birkhäuser, 1994). $\endgroup$ Dec 19, 2022 at 10:29

2 Answers 2


The solution to the mystery of how A. I. Petrov wrote a paper in 1964, when he was a student writing what appears to be his first paper in 1971, is that mathnet.ru has mistakenly taken his name to be A. I. Petrov, when in fact he is A. I. Perov, as you can see in the Wronskian paper. Moreover, Perov is still working at Voronezh State University, and I can send you his email address if you contact me at [email protected]. I don't know if he will help you, but the paper was part of a special collection of abstracts, so was not translated in the usual Uspekhi translations into English. The paper is pretty clearly translated using DeepL, so I don't think you really need a German or English reference.

  • $\begingroup$ thank you for the valuable information! I will first try the machine translation and, if that isn't sufficient, back to your offer of an email contact. $\endgroup$ Dec 17, 2022 at 18:25
  • $\begingroup$ Doklady or Uspekhi? These are different journals. $\endgroup$ Dec 18, 2022 at 14:49
  • $\begingroup$ @FedorPetrov: sorry, Uspekhi. $\endgroup$
    – Ben McKay
    Dec 18, 2022 at 16:36

A type of generalized Wronskian is used for the induction step in the proof of Roth's theorem in Diophantine approximation. There's a nice exposition in Schmidt's lecture note, for example, but undoubtedly it can be found in lots of other places. Roughly, the way that generalized Wronskians are used in the proof of Roth's theorem is to start with a polynomial $f(x_1,\ldots,x_r)$ and to construct a generalized Wronskian $W(f)$ that is the determinant of a matrix of partial derivatives (with controlled order) of $f$ having the property that $W(f)$ factors as $$ W\bigl(f(x_1,x_2,\ldots,x_r)\bigr) = g(x_1)h(x_2,\ldots,h_r). $$ This factorization is the key to doing an induction on the number of variables in the auxiliary polynomial.

Schmidt, Wolfgang M., Diophantine approximation, Lecture Notes in Mathematics. 785. Berlin-Heidelberg-New York: Springer-Verlag. 299 p.(1980). ZBL0421.10019.


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