Image Conjecture

Let be a field, and let be a commutative algebra over . Let be a polynomial ring, and let , ..., be a regular sequence in . The image conjecture asserts that the image of the -linear map given by is a Mathieu subspace of (van den Essen et al. 2011).

Taking and gives the complex-polynomial form in which and the image is .

When the field characteristic is zero, the image conjecture implies the vanishing conjecture, which is equivalent to the all-dimensional Jacobian conjecture (van den Essen et al. 2011). It is therefore false in some finite dimension, although this argument does not identify the smallest such dimension (Zhang 2026).