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03:00 PM
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04:00 PM

The Jacoboian Conjecture in dimension two states that if f = f (X, Y ) and g = g(X, Y ) are polynomials in two variables X and Y with real coefficients (or cofficients in any field k of characteristic zero) such that their Jacobian

J(f, g) = (∂f /∂X)(∂g/∂Y ) − (∂f /∂Y )(∂g/∂X)

is a nonzero constant then X and Y can be expressed as polynomials in f and g. In terms of rings, the conclusion can be stated as the equality k[f, g] = k[X, Y ].

There is a similar conjecture in dimension n.

The conjecture was formulated in 1939, and it is still open even for n = 2. The efforts of several mathematicians over the past decades have yielded only partial results.

We shall describe some of these partial results.

One of most significant contributions of (late) Professor Shriram Abhyankar is the following famous Abhyankar-Moh/Suzuki Epimorphism Theorem:

Theorem. Let k be a field of characteristic zero. Let F ∈ k[X, Y ] be such that .k[X, Y ]/(F ) ≈ k[T ]. Then F is a variable in k[X, Y ] i.e. ∃ G ∈ k[X, Y ] such that k[X, Y ] = k[F, G].

This result led to many important questions in affine geometry. To cite one (known as Abhyankar-Sathaye Epimorhism Problem):
Question. Let k be an algebraically closed field of characteristic zero. Let I be an ideal in the polynomial ring k[X₁ , X₂ , · · · , X_n] such that k[X₁ , X₂ , · · · , X_n]/I is a polynomial ring in m variables over k. Is it true that there exist F₁ , · · · , F_n−m ∈ I and G₁, · · · , G_m ∈ k[X₁ , X₂ , · · · , X_n ] such that

k[X₁ , X₂ , · · · , X_n ] = k[F₁, · · · , F_n−m, G₁ , · · · , G_m]?

In my talk, I will address this (and some other related questions) and report on the progress made so far.

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