FIRST SYZYGIES OF TORIC VARIETIES AND DIOPHANTINE EQUATIONS IN CONGRUENCE

Abstract

We compute the first syzygies of a subclass of lattice ideals by means of some abstract simplicial complexes. This subclass includes the ideals defining toric varieties. A finite check set containing the minimal first syzygy degrees is determined, and a singly-exponential bound for these degrees is explicited. Integer Programming techniques are used, precisely the Hilbert bases for diophantine equations in congruences.

^*Supported by Plan Propio de investigación de la Universidad de Sevilla (75403699-98-191). Fax: + 34 95 4556938; E-mail: [email protected]

^† Partially supported by Plan Propio de investigación de la Universidad de Sevilla (75403699-98-191) and Plan Propio de la Universidad de Cádiz. Fax: + 34 956 345104; E-mail: [email protected]

Keywords:

• Toric varieties
• Syzygies
• Simplicial complexes
• Diophantine equations
• 1991 Mathematics Subject Classification: 13DO2, 13P1O

Acknowledgments

Notes

^*Supported by Plan Propio de investigación de la Universidad de Sevilla (75403699-98-191). Fax: + 34 95 4556938; E-mail: [email protected]

^† Partially supported by Plan Propio de investigación de la Universidad de Sevilla (75403699-98-191) and Plan Propio de la Universidad de Cádiz. Fax: + 34 956 345104; E-mail: [email protected]