代数方程可解性及解的结构表示是代数学中的重要研究内容，在数学本身及系统科学、工程、控制论、物理、计算机以及信息等领域都有着重要的应用。无论是代数学本身的发展，还是实际问题的解决，都需要深入研究代数方程。..本项目是非交换代数上代数方程与系统控制及量子计算紧密结合的国际前沿性课题。旨在研究若干非交换代数上代数方程和代数方程群的可解性、通解、各种解的结构表示及其在系统控制和量子计算中的应用。..本项目的实施将有助于非交换代数上代数方程及其在控制和量子计算中实际应用的深入研究，从而推动代数学的进一步发展和应用。

The researches on the solvability conditions and the structural representations of solutions to algebraic equations have been one of the significant topics in algebra for a long time. Nowadays as one important part of contemporary mathematics, algebraic equations are widely and heavily used in many areas such as engineering, system and control theory, computer science, physics, and information science. No matter concerning the development of algebra or solving practical problems, further studying on algebraic equations is essential. ..This proposal focuses on an international leading area: algebraic equations over non-commutative algebras which have very closed relations with system and control theory and quantum computing. The main questions include the solvability conditions, general solutions, structural representations of the solutions to some algebraic equations over non-commutative algebra. Moreover, we will explore their applications in system and control theory and quantum computing...The implementation of this project will contribute to the thorough research of the algebra equations and their practical applications in system and control theory and quantum computing, so as to promote the further development of algebra and its applications.