We study polynomial systems with prescribed monomial supports in the Cox rings of toric varieties built from complete polyhedral fans. We present combinatorial formulas for the dimensions of their associated subvarieties under genericity assumptions on the coefficients of the polynomials. Using these formulas, we identify at which degrees generic systems in polytopal algebras form regular sequences. Our motivation comes from sparse elimination theory, where knowing the expected dimension of these subvarieties leads to specialized algorithms and to large speed-ups for solving sparse polynomial systems. As a special case, we classify the degrees at which regular sequences defined by weighted homogeneous polynomials can be found, answering an open question in the Gr\"obner bases literature. We also show that deciding whether a sparse system is generically a regular sequence in a polytopal algebra is hard from the point of view of theoretical computational complexity.

翻译：我们研究了由完全多面体扇构建的环面簇的Cox环中具有给定单项式支撑集的多项式系统。在多项式系数的一般性假设下，我们给出了这些多项式所关联子簇维数的组合公式。利用这些公式，我们确定了多面体代数中一般系统在何种次数下构成正则序列。我们的动机源于稀疏消元理论——了解这些子簇的预期维数可催生专用算法，并大幅加速稀疏多项式系统的求解。作为特例，我们分类了加权齐次多项式定义的正则序列可被发现的次数，回答了Gröbner基文献中的一个开放问题。我们还证明，从理论计算复杂性的角度来看，判断稀疏系统在多面体代数中是否一般为正则序列是困难的。