Abstract
Multivariate Hermitian quadratic forms play an important role in the real quantifier elimination algorithm based on the computation of comprehensive Gröbner systems introduced by V. Weispfenning and further improved by us. Our algorithm needs the computation of a certain type of saturation ideal in a parametric polynomial ring. In this paper, we study multivariate Hermitian quadratic forms in more detail and show several facts which have special importance in a parametric polynomial ring. Our results enable us to have an efficient method to compute the saturation ideal, which brings us a drastic improvement of our real quantifier elimination software.
Similar content being viewed by others
References
Arai, N.H., Matsuzaki, T., Iwane, H., Anai, H.: Mathematics by machine. In: Proceedings of International Symposium on Symbolic and Algebraic Computation, pp. 1–8, ACM-Press (2014)
Becker, E., Wörmann, T.: On the trace formula for quadratic forms. In: Proceedings of Recent Advances in Real Algebraic Geometry and Quadratic Forms, Contemporary Mathematics, vol. 155, pp. 271–291. American Mathematical Society (1994)
Brown, C., et al.: QEPCAD. https://urldefense.proofpoint.com/v2/url?u=https-3A__www.usna.edu_CS_qepcadweb_B_QEPCAD.html&d=DwICAg&c=vh6FgFnduejNhPPD0fl_yRaSfZy8CWbWnIf4XJhSqx8&r=UpNksRRkQEKvUbblp9QRYTuemGLwQgpW1U7iMCFPZ8k&m=mWR4kwzCQJx99sfByc_Wd_NtOKrvA94px0mf4a_qj_8&s=fs72Aj2MHHkrUbt5MXeOF0B3iyfTQkLWYjPboUB-RSA&e=. Accessed 8 Oct 2017
Fukasaku, R.: 2016 Version of CGSQE Package. https://urldefense.proofpoint.com/v2/url?u=http-3A__www.rs.tus.ac.jp_fukasaku_software_CGSQE-2D20160509_&d=DwICAg&c=vh6FgFnduejNhPPD0fl_yRaSfZy8CWbWnIf4XJhSqx8&r=UpNksRRkQEKvUbblp9QRYTuemGLwQgpW1U7iMCFPZ8k&m=mWR4kwzCQJx99sfByc_Wd_NtOKrvA94px0mf4a_qj_8&s=ExGbD7jMK0fzhZjaF_9DQ2vYfqv6alRkPjcYhRzkADk&e=. Accessed 8 Oct 2017
Fukasaku, R.: 2017 Version of CGSQE Package. https://urldefense.proofpoint.com/v2/url?u=http-3A__www.rs.tus.ac.jp_fukasaku_software_CGSQE-2D2017_&d=DwICAg&c=vh6FgFnduejNhPPD0fl_yRaSfZy8CWbWnIf4XJhSqx8&r=UpNksRRkQEKvUbblp9QRYTuemGLwQgpW1U7iMCFPZ8k&m=mWR4kwzCQJx99sfByc_Wd_NtOKrvA94px0mf4a_qj_8&s=E-8Gmc7etlgpy1LN6yFA1qKUNLLB65V2URmVednn2Rg&e=. Accessed 8 Oct 2017
Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Proceedings of Automata Theory and Formal Languages, LNCS vol. 33, pp. 134–183. Springer, Berlin (1975)
Fukasaku, R.: QE software based on comprehensive Gröbner systems. In: Proceedings of Mathematical Software—ICMS 2014—4th International Congress, LNCS vol. 8592, pp. 512–517. Springer, Berlin (2014)
Fukasaku, R., Iwane, H., Sato, Y: Real quantifier elimination by computation of comprehensive Gröbner systems. In: Proceedings of International Symposium on Symbolic and Algebraic Computation, pp. 173–180. ACM-Press (2015)
Fukasaku, R., Iwane, H., Sato, Y: On the Implementation of CGS Real QE. In: Proceedings of Mathematical Software—ICMS 2016—5th International Conference, LNCS vol. 9725, pp. 165–172. Springer, Berlin (2016)
Iwane, H.: SyNRAC Package. https://urldefense.proofpoint.com/v2/url?u=http-3A__www.fujitsu.com_jp_group_labs_en_resources_tech_freeware_synrac_&d=DwICAg&c=vh6FgFnduejNhPPD0fl_yRaSfZy8CWbWnIf4XJhSqx8&r=UpNksRRkQEKvUbblp9QRYTuemGLwQgpW1U7iMCFPZ8k&m=mWR4kwzCQJx99sfByc_Wd_NtOKrvA94px0mf4a_qj_8&s=ohEYZdyMb9fiZ7ajsP8WnI6QvqhisstydbZQtD1hGkM&e=. Accessed 8 Oct 2017
Kapur, D., Sun, Y., Wang, D.: A new algorithm for computing comprehensive Gröbner systems. In: Proceedings of International Symposium on Symbolic and Algebraic Computation, pp. 29–36. ACM-Press (2010)
Kurata, Y.: Improving Suzuki–Sato’s CGS algorithm by using stability of Gröbner bases and basic manipulations for efficient implementation. Commun. Jpn. Soc. Symb. Algebr. Comput. 1, 39–66 (2011)
Maza, M.-M. et al.: RegularChains Package. https://urldefense.proofpoint.com/v2/url?u=http-3A__www.regularchains.org_&d=DwICAg&c=vh6FgFnduejNhPPD0fl_yRaSfZy8CWbWnIf4XJhSqx8&r=UpNksRRkQEKvUbblp9QRYTuemGLwQgpW1U7iMCFPZ8k&m=mWR4kwzCQJx99sfByc_Wd_NtOKrvA94px0mf4a_qj_8&s=NJ2uBtKt_hCxEF7lQWZjZVvTVQSMTWIHFjDD-FsbidA&e=. Accessed 8 Oct 2017
Nabeshima, K.: A speed-up of the algorithm for computing comprehensive Gröbner systems. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation, pp. 299–306. ACM-Press (2007)
Nabeshima, K.: Stability conditions of monomial bases and comprehensive Gröbner systems. In: Proceedings of Computer Algebra in Scientific Computing, LNCS vol. 7442, pp. 248–259. Springer, Berlin (2012)
Pedersen, P., Roy, M.-F., Szpirglas, A.: Counting real zeroes in the multivariate case. In: Proceedings of Effective Methods in Algebraic Geometry, Progress in Mathematics vol. 109, pp. 203–224. Springer, Berlin (1993)
Sato, S., Fukasaku, R., Sekigawa, H.: On continuity of the roots of a parametric zero dimensional multivariate polynomial ideal. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation, pp. 359–365. ACM-Press (2018)
Strzebonski, A.: Reduce of Mathematica. https://urldefense.proofpoint.com/v2/url?u=https-3A__reference.wolfram.com_language_ref_Reduce.html&d=DwICAg&c=vh6FgFnduejNhPPD0fl_yRaSfZy8CWbWnIf4XJhSqx8&r=UpNksRRkQEKvUbblp9QRYTuemGLwQgpW1U7iMCFPZ8k&m=mWR4kwzCQJx99sfByc_Wd_NtOKrvA94px0mf4a_qj_8&s=4EgFsOWDHosP2kivGNaEgtF21HCLwzXqfkkgmrzY7SI&e=. Accessed 8 Oct 2017
Strzebonski, A.: Resolve of Mathematica. https://urldefense.proofpoint.com/v2/url?u=https-3A__reference.wolfram.com_language_ref_Resolve.html&d=DwICAg&c=vh6FgFnduejNhPPD0fl_yRaSfZy8CWbWnIf4XJhSqx8&r=UpNksRRkQEKvUbblp9QRYTuemGLwQgpW1U7iMCFPZ8k&m=mWR4kwzCQJx99sfByc_Wd_NtOKrvA94px0mf4a_qj_8&s=f-nt87scWnkm1Ct-Hg0T9_eYCwKRzyOsZBx-bnIlD10&e=. Accessed 8 Oct 2017
Strzebonski, A.: Solving systems of strict polynomial inequalities. J. Symb. Comput. 29(3), 471–480 (2000)
Sturm, T. et al.: Redlog package. https://urldefense.proofpoint.com/v2/url?u=http-3A__www.redlog.eu_&d=DwICAg&c=vh6FgFnduejNhPPD0fl_yRaSfZy8CWbWnIf4XJhSqx8&r=UpNksRRkQEKvUbblp9QRYTuemGLwQgpW1U7iMCFPZ8k&m=mWR4kwzCQJx99sfByc_Wd_NtOKrvA94px0mf4a_qj_8&s=IYEAxvdgb7Rlopn5s55MJGTASSLXMa0Tr395COkW5Ww&e=. Accessed 8 Oct 2017
Suzuki, A., Sato, Y.: A simple algorithm to compute comprehensive Gröbner bases using Gröbner bases. In: Proceedings of International Symposium on Symbolic and Algebraic Computation, pp. 326–331. ACM-Press (2006)
Weispfenning, V.: A new approach to quantifier elimination for real algebra. In: Quantifier Elimination and Cylindrical Algebraic Decomposition, pp. 376–392. Springer, Berlin (1998)
Acknowledgements
This work was partially supported by JSPS KAKENHI Grant Numbers 17K12642 and 18K03426.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fukasaku, R., Iwane, H. & Sato, Y. On Multivariate Hermitian Quadratic Forms. Math.Comput.Sci. 13, 79–93 (2019). https://doi.org/10.1007/s11786-018-0387-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11786-018-0387-8