[Paper Review] Heights of varieties in multiprojective spaces and arithmetic Nullstellensatze

Carlos D’Andrea, Térésa Krick+1 more|arXiv (Cornell University)|Mar 23, 2011

Polynomial and algebraic computation17 references4 citations

TL;DR

This paper establishes effective bounds for the degree and height of polynomials in arithmetic Nullstellensätze over varieties in multiprojective spaces, using arithmetic intersection theory and canonical mixed heights. It extends Jelonek's geometric results to the arithmetic setting, providing explicit degree and height estimates for Bézout identities on varieties, with applications to implicitization and effective algebraic geometry over number fields.

ABSTRACT

We present bounds for the degree and the height of the polynomials arising in some central problems in effective algebraic geometry including the implicitation of rational maps and the effective Nullstellensatz over a variety. Our treatment is based on arithmetic intersection theory in products of projective spaces and extends to the arithmetic setting constructions and results due to Jelonek. A key role is played by the notion of canonical mixed height of a multiprojective variety. We study this notion from the point of view of resultant theory and establish some of its basic properties, including its behavior with respect to intersections, projections and products. We obtain analogous results for the function field case, including a parametric Nullstellensatz.

Motivation & Objective

• To develop effective bounds for the degree and height of polynomials in Bézout identities over algebraic varieties in multiprojective spaces.
• To extend geometric results of Jelonek on effective Nullstellensätze to the arithmetic setting using arithmetic intersection theory.
• To introduce and study the canonical mixed height of cycles in products of projective spaces, particularly over Q and function fields.
• To establish parametric and strong arithmetic Nullstellensätze with explicit degree and height estimates.
• To apply these bounds to problems in implicitization and effective algebraic geometry over number fields.

Proposed method

• The authors define and analyze the canonical mixed height of cycles in multiprojective spaces using tools from arithmetic intersection theory and resultant theory.
• They establish bounds on the height of the implicit equation via elimination theory and the use of eliminants and resultants in multiprojective geometry.
• The method involves lifting polynomials to function fields, applying parametric Nullstellensätze, and extracting integral identities via coefficient extraction.
• Key technical tools include the use of mixed degrees, measures of complex polynomials, and height estimates via logarithmic norms of coefficients.
• The approach combines algebraic geometry with arithmetic geometry, particularly leveraging the canonical height of varieties and degree bounds via resultants.
• The construction relies on a parametrization of coefficients and monomial support to control height growth in Bézout identities.

Experimental results

Research questions

• RQ1 What are effective degree and height bounds for polynomials in a Bézout identity that vanish on a given variety over Q?
• RQ2 How does the canonical mixed height of a cycle in a product of projective spaces relate to its geometric and arithmetic complexity?
• RQ3 Can the parametric Nullstellensatz be used to derive effective bounds for families of polynomials over varieties with varying coefficients?
• RQ4 What is the behavior of the height of the implicit equation under rational maps in multiprojective spaces?
• RQ5 How do mixed degrees and resultants in multiprojective geometry control the complexity of effective Nullstellensätze?

Key findings

• For a variety $ V \subset \mathbb{A}^n(\overline{\mathbb{Q}}) $ of pure dimension $ r $, and polynomials $ f_1, \dots, f_s \in \mathbb{Z}[x_1, \dots, x_n] $ with $ s \leq r+1 $ and no common zeros on $ V $, there exists a Bézout identity $ \alpha = \sum g_i f_i $ on $ V $ with $ \deg(g_i f_i) \leq \left( \prod_{j=1}^s d_j \right) \deg(V) $, where $ d_j = \deg(f_j) $.
• The height of $ \alpha $ and of each $ g_i $ satisfies $ \operatorname{h}(\alpha), \operatorname{h}(g_i) + \operatorname{h}(f_i) \leq \left( d_s \prod_{j=1}^r d_j \right) \left( \widehat{h}(V) + \deg(V) \left( \frac{h_s}{d_s} + \sum_{\ell=1}^r \frac{h}{d_\ell} + 3r\log(s-r) + (6r+8)\log(n+3) \right) \right) $, with $ h_j = \operatorname{h}(f_j) $.
• The paper proves a parametric Nullstellensatz over function fields, showing that for a family of polynomials with coefficients in $ \mathbb{Z}[\mathbf{v}] $, the Bézout identity can be lifted with controlled degree and height in terms of the support and degree of the coefficients.
• For the arithmetic strong Nullstellensatz, the paper derives bounds for $ \alpha $ and $ g_i $ in the identity $ \alpha g^\mu = \sum g_i f_i $ on $ V $, with explicit height control involving $ \widehat{h}(V) $, $ \deg(V) $, and logarithmic terms in $ n $, $ d_j $, and $ s $.
• The canonical mixed height of a cycle is shown to behave well under projections, intersections, and products, and is characterized via resultant theory and mixed degrees.
• The height of the implicit equation in rational maps is bounded by a function of the degrees and heights of the input maps, with explicit dependence on the number of variables and the size of the coefficient support.