پایه گربنر ایدهآلهای دترمینانی.

Introduction Determinantal ideals are one of important topics in Algebraic Geometry and Commutative Algebra. There are several examples of varieties as rational normal scrolls which their defining ideals are generated by minors of a matrix. In general, Grobner bases for an ideals helps us to correspond a monomial ideal to the main ideal with the same invariants in some senses. In this paper, we compute a Grobner bases for an ideal generated by 2-minors of a 2×n matrix of monomials. Main results Let S = k[x₁, ..., x_n] be the polynomial ring over a field k. Let for 1 ≤ i ≤ n, X_i = {x_il, 1 ≤ l ≤ s} for some positive integer s. Let M= [ m₁₁(X₁ ) ⋯ m_1n (X_n) ⋮ ... ⋮ m_t1 (X₁) ⋯ m_tn(X_n)] be a matrix such that m_ij(Xj) is a monomial of indeterminates in Xj. A t-minor [r₁ r₂ ... r_t | c₁ c₂ ... c_t] of M is determinant of the submatrix with rows r₁ r₂ ... r_t and columns c₁ c₂ ... c_t. Let It (M) be the ideal generated by all t-minors of M. Let ≤ be a monomial order and the following conditions are satisfied: 1. Each column is decreasing from top to bottom. 2. Each monomial in a column is greater that each monomial in a column in the right. 3. Each two monomials are different. With these assumptions, we have the following theorem. Theorem. For t = 2, set of all 2-minors of M is a Grobner bases for I₂ (M). Proposition. Height of I_t (M) is equal to n-t+1. Conclusion A 2 by n matrix of monomials appears in some topics of algebraic combinatorics and if we omit each one of the above conditions, the theorem might be false. [ABSTRACT FROM AUTHOR]