We give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties. We also extend the Jacobian dual criterion to the multi-graded setting. Our approach is based on the study of blow-up algebras, including syzygies, of the ideal generated by the defining polynomials of the rational map. A key ingredient is a new algebra that we call the "saturated special fiber ring", which turns out to be a fundamental tool to analyze the degree of a rational map. We also provide a very effective birationality criterion and a complete description of the equations of the associated Rees algebra of a particular class of plane rational maps.

中文翻译：

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多级有理图的度和二元性

我们给出了多级有理图的程度的公式和有效的尖锐边界，并根据其代数和几何性质，为双合性提供了一些有效且可计算的标准。我们还将雅可比对偶准则扩展到多级设置。我们的方法是基于对由有理图的定义多项式生成的理想的爆炸代数（包括sysygie）进行研究的。关键要素是一个新的代数，我们称之为“饱和特殊纤维环”，它是分析有理图的程度的基本工具。我们还提供了一个非常有效的双合性准则，以及一类特定平面有理图的相关Rees代数方程的完整描述。