On the morning of 28 March 2026, at VNUHCM–University of Science (HCMUS), researcher Trần Nam Sơn (class of 2022, specialising in Algebra and Number Theory) successfully defended a doctoral thesis at the institutional level. The research is entitled: “Images of polynomials on certain classes of algebras”.

The thesis was conducted under the academic supervision of Professor Mai Hoàng Biên. The study focuses on investigating the images of polynomials when evaluated on non-commutative algebras, with a particular emphasis on generalised non-commutative polynomials and the connection to the L’vov–Kaplansky Conjecture and the Mesyan Conjecture.

The scope of the research involves examining the behaviour of polynomials across various classes of algebras, such as division algebras, matrix algebras over division rings, quaternion algebras, and other specialised algebraic structures. On this basis, the thesis considers instances where the image of a polynomial may cover the entire algebra or form characteristic subspaces.

Furthermore, the research investigates the representation of elements as products of squares or multiplicative commutators. These findings contribute to establishing links with problems in linear group theory and Waring-type problems within a non-commutative context.

Regarding the results, the thesis established necessary and sufficient conditions concerning algebraic structures through the images of polynomials. The researcher successfully described the images of multilinear polynomials on quaternion algebras and clarified specific cases where the polynomial image coincides with the entire algebra. Additionally, the study proved extended results related to Kursov’s Theorem and analysed the relationship between multiplicative commutators and squares in several classes of algebras.

The Examination Board evaluated the thesis as being of high quality, possessing significant scientific merit and professional value within the fields of Algebra and Number Theory. The researcher demonstrated a capacity for independent research, fully satisfying the requirements for a doctoral thesis. The findings are reliable, original, and do not duplicate previously published works, thereby contributing to the clarification of advanced problems within the field of study.