Ghiloni, Riccardo (2004) *Explicit equations and bounds for the Nakai-Nishimura-Dubois-Efroymson dimension theorem.* UNSPECIFIED. (Unpublished)

## Abstract

The Nakai-Nishimura-Dubois-Efroymson dimension theorem asserts the following: "let R be an algebraically closed field or a real closed field, let X be an irreducible algebraic subset of Rn and let Y be an algebraic subset of X of codimention s>=2 (not necessarily irreducible). Then, there is an irreducible algebraic subset W of X of codimention 1 containing Y". In this paper, making use of an elementary construction, we improve this result giving explicit polynomial equations for W. Moreover, denoting by R the algebraic closure of R and embedding canonically W into projective space Pn(R), we obtain explicit upper bounds for the degree and the geometric genus of the Zariski closure of W in Pn(R). In future papers, we will use these bounds in the study of morphism space between algebraic varieties over real closed fields.

Item Type: | Departmental Technical Report |

Department or Research center: | Mathematics |

Subjects: | Q Science > QA Mathematics > QA152 Algebra |

Uncontrolled Keywords: | Dimension theorems, Irreducible algebraic subvarieties, Upper bounds for the degree of algebraic varieties, Upper bounds for the geometric genus of algebraic varieties. |

Report Number: | UTM 661, February 2004 |

Repository staff approval on: | 17 May 2005 |
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