We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for hypergeometric systems  H A ( β ) H_A(\beta ) arising from a d × n d \times n integer matrix  A A and a parameter β ∈ C d \beta \in \mathbb {C}^d . To do so we introduce an Euler–Koszul functor for hypergeometric families over  C d \mathbb {C}^d , whose homology generalizes the notion of a hypergeometric system, and we prove a homology isomorphism with our general homological construction above. We show that a parameter β ∈ C d \beta \in \mathbb {C}^d is rank-jumping for H A ( β ) H_A(\beta ) if and only if β \beta lies in the Zariski closure of the set of C d \mathbb {C}^d -graded degrees  α \alpha where the local cohomology ⨁ i > d H m i ( C [ N A ] ) α \bigoplus _{i > d} H^i_\mathfrak m(\mathbb {C}[\mathbb {N} A])_\alpha of the semigroup ring C [ N A ] \mathbb {C}[\mathbb {N} A] supported at its maximal graded ideal  m \mathfrak m is nonzero. Consequently, H A ( β ) H_A(\beta ) has no rank-jumps over  C d \mathbb {C}^d if and only if C [ N A ] \mathbb {C}[\mathbb {N} A] is Cohen–Macaulay of dimension  d d .

We study quotients of the Weyl algebra by left ideals whose generators consist of an arbitrary Z d -graded binomial ideal I in C[∂ 1 , . . . , ∂ n ] along with Euler operators defined by the grading and a parameter β ∈ C d . We determine the parameters β for which these D-modules (i) are holonomic (equivalently, regular holonomic, when I is standard-graded); (ii) decompose as direct sums indexed by the primary components of I; and (iii) have holonomic rank greater than the rank for generic β. In each of these three cases, the parameters in question are precisely those outside of a certain explicitly described affine subspace arrangement in C d . In the special case of Horn hypergeometric D-modules, when I is a lattice basis ideal, we furthermore compute the generic holonomic rank combinatorially and write down a basis of solutions in terms of associated A-hypergeometric functions. This study relies fundamentally on the explicit lattice point description of the primary components of an arbitrary binomial ideal in characteristic zero, which we derive in our companion article [DMM08]. CONTENTSTo construct a binomial D-module, the starting point is an integer matrix A, about which we wish to be consistent throughout.Convention 1.2. A = (a ij ) ∈ Z d×n denotes an integer d×n matrix of rank d whose columns a 1 , . . . , a n all lie in a single open linear half-space of R d ; equivalently, the cone generated by the columns of A is pointed (contains no lines), and all of the a i are nonzero. We also assume that ZA = Z d ; that is, the columns of A span Z d as a lattice.The reformulation of Horn systems in Section 1.4 proceeds by a change of variables, so we will use x = x 1 , . . . , x n and ∂ = ∂ 1 , . . . , ∂ n (where ∂ i = ∂ x i = ∂/∂x i ), instead of z 1 , . . . , z m and ∂ z 1 , . . . , ∂ zm , whenever we work in the binomial setting. The matrix A induces a Z dgrading of the polynomial ring C[∂ 1 , . . . , ∂ n ] = C[∂], which we call the A-grading, byis A-graded if it is generated by elements that are homogeneous for the A-grading. For example, a binomial ideal is generated by binomials ∂ u −λ∂ v , where u, v ∈ Z n are column vectors and λ ∈ C; such an ideal is A-graded precisely when it is generated by binomials ∂ u − λ∂ v each of which satisfies either Au = Av or λ = 0 (in particular, monomials are allowed as generators of binomial ideals). The hypotheses on A mean that the A-grading is a positive Z d -grading [MS05, Chapter 8].The Weyl algebra D = D n of linear partial differential operators, written with the variables x and ∂, is also naturally A-graded by additionally setting deg(x i ) = a i . Consequently, the Euler operators in our next definition are A-homogeneous of degree 0. Definition 1.3. For each i ∈ {1, . . . , d}, the i th Euler operator isGiven a vector β ∈ C d , we write E − β for the sequence E 1 − β 1 , . . . , E d − β d . (The dependence of the Euler operators E i on the matrix A is suppressed from the notation.) Lemma 7.10. If β is A J -nonresonant, then for any γ ∈ N J , and for all torus trans...

We undertake the study of bivariate Horn systems for generic parameters. We prove that these hypergeometric systems are holonomic, and we provide an explicit formula for their holonomic rank as well as bases of their spaces of complex holomorphic solutions. We also obtain analogous results for the generalized hypergeometric systems arising from lattices of any rank.

ABSTRACT. An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristic-free combinatorial description of certain primary components of binomial ideals in affine semigroup rings, namely those that are associated to faces of the semigroup. These results are intimately connected to hypergeometric differential equations in several variables.

Abstract.We study the solutions of irregular A-hypergeometric systems that are constructed from Gröbner degenerations with respect to generic positive weight vectors. These are formal logarithmic Puiseux series that belong to explicitly described Nilsson rings, and are therefore called (formal) Nilsson series. When the weight vector is a perturbation of (1, . . . , 1), these series converge and provide a basis for the (multivalued) holomorphic hypergeometric functions in a specific open subset of C n . Our results are more explicit when the parameters are generic or when the solutions studied are logarithm-free. We also give an alternative proof of a result of Schulze and Walther that inhomogeneous A-hypergeometric systems have irregular singularities.

a b s t r a c tAs the highest precision devices of celestial navigation system [1], star sensors have been getting more and more attention in recent years. In which the star image positioning and recognition is the key technology of CNS, while the extraction of stars from star maps is the first step. By the background noise, there are some error extractions when traditional methods are used, which can even lead to the failure of star map matching. To solve this problem, a denoising method based on overcomplete sparse representation is presented in this paper. This method uses the adaptive sparse decomposition of star map in the redundant dictionary to process the threshold, as a result, the reliability of star extraction is improved. The experimental results show that the correct rate of this method that extracting star after reducing background noise of star map is close to 100%.

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ABSTRACT. Without any restrictions on the base field, we compute the hull and prove a conjecture of Eisenbud and Sturmfels giving an unmixed decomposition of a cellular binomial ideal. Over an algebraically closed field, we further obtain an explicit (but not necessarily minimal) primary decomposition of such an ideal.