As an effective use of the CAS (computer algebra system) in the mathematics education, we can perform mathematics experiment and can do heuristic learning. One of the roles expected to use of CAS in mathematics learning is to be able to carry out experimental mathematics to make heuristically by myself. We introduce a trial (mathematical search) to practice such an activity in university mathematics. We can obtain degeneracy conditions of singularities by using Gr\"obner basis. In classification of singularities, the defining equations of certain types are polynomials with parametric coefficients. In general, the Gr\"obner basis generated by such polynomials depends on the values of the parameters, and those calculations are not easy. A structure of zero set defined by polynomials with parameters changes by the values of the parameters. This is the learning problem that is important in the university mathematics. As one of such teaching materials, the defining equation of singularity with parameters is suitable. We show various examples of the calculations to students and teach these calculations. When students learn those calculations as algorithm, an interesting subject is preferable, mathematically. The singularity on the algebraic curve can arouse the interest of the students. We can make the learning of the algebraic curve with parameters more interesting in university mathematics. This trial is significant to show utility of the mathematics by using computer and to clarify the pleasure of the calculations to students. Then, the calculation based on Gr\"obner basis theory is necessary. If they calculate thoughtlessly, they will not obtain significant results. The mathematical result to show in this paper is still an experiment stage on the way. We show these search activities as the material of the experiment in university mathematics in this paper.