Using gerund: "applying" or "on applying"?

In a mathematical paper my co-author wrote: "On applying s to the coefficients of the polynomials defining our variety X, we obtain a new variety sX".

The anonymous referee suggested a correction: "Applying s to the coefficients ...", without On.

Which version is better? We want to say that after we apply s to X, we obtain a new variety sX.

Maybe "By applying s ..."?

Solution 1:

"PREP applying s to the coefficients of the polynomials defining our variety X, we obtain a new variety sX".

(a) PREP = By

This highlights the actual method used, emphasising this and to some degree downplaying the result.

(b) PREP = On

This connotes either a slight degree of mundaneness or this being one in a long line of operations, or an unfortunate hint of an unexpected result.

(c) zero PREP

This is the unmarked version. It is businesslike. The lack of a preposition does not mask the sequentiality (starting element) .>. (transformation) .>. (image).

Solution 2:

On applying, as you observe is a gerund, and the critic wants to know what this verbal-noun phrase, in this sentence, is in apposition to. It is sort of adverbial with reference to obtain. It works and everyone understands what you mean, but it is clumsy. (paraphrase: When there is an application of...)

but

Applying is a verbal adjective, a present participle in old money, and it clearly refers to ", we", which is clearly the subject of the main clause. The syntax is simpler and the referent is clear. (paraphrase: When we apply... , we...)