Algorithmic algebra and symbolic computation have important scientific significance and application value in mathematical theory and engineering calculations. Some problems in many mathematical and engineering fields, such as algebraic geometry, computer algebra, algebraic topology, circuit analysis, multi-dimensional control, signal processing, multi-dimensional systems, can be transformed into problems of multivariate polynomial matrix. In 1955, J.P. Serre put forward the famous Serre conjecture: any finitely generated projective module in a polynomial ring over a field must be a free module, which is equivalent to that any ZLP matrix can be embedded into a invertible matrix. The embedding problem of a ZLP matrix A is equivalent to finding a right identity matrix of A, i.e., finding the invertible matrix U such that AU=($E_r$, 0). Logar and Fabianska gaved Quillen-Suslin algorithm involved calculating polynomial zeros, but there is no general algorithm for calculating polynomial zeros. The embedding of ZLP polynomial matrix can be transformed into the embedding problem of unimodular rows by recursive method. This paper mainly investigate the existence of the minimal syzygy module of ZLP polynomial matrix, and demonstrate that the minimal syzygy module has structural properties similar to the fundamental solution system of homogeneous linear equations in linear algebra. We proved that the minimal syzygy module of ZLP matrix is free, and the embedding problem of unimodular vector is studied by avoiding finding the zeros of polynomials. The generator set of the syzygy module of unimodular vector is given in formulaic form, and the invertible matrix of unimodular vector embedded in it is further given in several cases.