Jose Abel Castellanos Joo

Jose Abel Castellanos Joo

Ph.D. in Computer Science

SatQMCert

The code is available here

The package SatQMCert implements the algorithm in (Castellanos-Joo and Kapur 2026) to compute certificates of elements in saturated quadratic module. The algorithm is fully symbolic in the sense that it does not depend on numerical algorithm to obtain certificates.

Dependencies

  • BasicLemma: This maple package implements an algorithm of Lemma 2.1 in (Kuhlmann et al. 2005).
  • StrictlyPositiveCert: This maple package implements the algorithm Algorithm 4 proposed in (Shang et al. 2025) which is used to compute certificates of strictly positive polynomial on Archimedean quadratic modules.
  • (Optional)RealCertify: RealCertify is a maple package that also computes certificates of strictly positive polynomial over Archimedean quadratic modules. It can be used instead of StrictlyPositiveCert and it is used in some comparision benchmarks.

Installation

Use the make command at the root of the project to produce the maple library file SatQMCert.mla. Then, load the SatQMCert package by adding the following two lines at the beginning of your maple file:

libname := libname, "<PATH-TO-SATQMCERT>/SatQMCert.mla":
with(SatQMCert):

Usage and examples

The command certInQM has the following inputs:

  • f: Denotes the input polynomial
  • nat: Denotes the set of natural generators associated to the original generators
  • a_0: Denotes the minimum point in the semialgebraic set associated to the original generators
  • b_k: Denotes the maximum point in the semialgebraic set associated to the original generators
  • x: Denotes the free variable of the ring of univariate polynomial

For examples, check the directory tests.

References

Kuhlmann, S., M. Marshall, and N. Schwartz. 2005. Positivity, Sums of Squares and the Multi-Dimensional Moment Problem II. 5 (4): 583–606. https://doi.org/10.1515/advg.2005.5.4.583.

Shang, Weifeng, Chenqi Mou, Jose Abel Castellanos Joo, and Deepak Kapur. 2025. Computing Certificates of Strictly Positive Polynomials in Archimedean Quadratic Modules. https://arxiv.org/abs/2503.11119.

Castellanos-Joo, Jose Abel, and Deepak Kapur. 2026. Computing Certificates in Archimedean Univariate Saturated Quadratic Modules. https://arxiv.org/abs/2605.18980.