On the Maximal Rank Conjecture for Line Bundles of Extremal Degree

We propose a new method, using deformation theory, to study the maximal rank conjecture. For line bundles of extremal degree, which can be viewed as the first case to test the conjecture, we prove that maximal rank conjecture holds by our new method

Some results on the generic vanishing of Koszul cohomology via deformation theory

We study the deformation-obstruction theory of Koszul cohomology groups of $g^r_d$'s on singular nodal curves. We compute the obstruction classes for Koszul cohomology classes on singular curves to deform to a smooth one. In the case the obstructions are nontrivial, we obtain some partial results for generic vanishing of Koszul cohomology groups.Comment: 25 pages, to appear in Pacific J. Mat

On Supports of Sums of Nonnegative Circuit Polynomials

In this paper, we prove that every SONC polynomial decomposes into a sum of nonnegative circuit polynomials with the same support, which reveals the advantage of SONC decompositions for certifying nonnegativity of sparse polynomials compared with the classical SOS decompositions. By virtue of this fact, we can decide $f\in$ SONC through relative entropy programming more efficiently.Comment: 7 pages. arXiv admin note: substantial text overlap with arXiv:1804.0945

Difference Index of Quasi-regular Difference Algebraic Systems

This paper is devoted to studying difference indices of quasi-regular difference algebraic systems. We give the definition of difference indices through a family of pseudo-Jacobian matrices. Some properties of difference indices are proved. In particular, a Jacobi-type upper bound for the sum of the order and the difference index is given. As applications, an upper bound of the Hilbert-Levin regularity and an upper bound of the order for difference ideal membership problem are deduced.Comment: 11 page

On Systems of Polynomials with At Least One Positive Real Zero

In this paper, we prove several theorems on systems of polynomials with at least one positive real zero, which can be viewed as a partial generalization of Birch's theorem. The method is based on the theory of conceive polynomials. Moreover, a class of polynomials attaining their global minimums are given, which is useful in polynomial optimization.Comment: 10 page

Monomial Difference Ideals

In this paper, basic properties of monomial difference ideals are studied. We prove the finitely generated property of well-mixed difference ideals generated by monomials. Furthermore, a finite prime decomposition of radical well-mixed monomial difference ideals is given. As a consequence, we prove that every strictly ascending chain of radical well-mixed monomial difference ideals in a difference polynomial ring is finite, which answers a question raised by E. Hrushovski in the monomial case. Moreover, the Alexander Duality for monomial ideals is generalized to the monomial case.Comment: 16 page

Difference Index of Quasi-Prime Difference Algebraic Systems

This paper is devoted to studying difference indices of quasi-prime difference algebraic systems. We define the quasi dimension polynomial of a quasi-prime difference algebraic system. Based on this, we give the definition of the difference index of a quasi-prime difference algebraic system through a family of pseudo-Jacobian matrices. Some properties of difference indices are proved. In particular, an upper bound of difference indices is given. As applications, an upper bound of the Hilbert-Levin regularity and an upper bound of orders for difference ideal membership problem are deduced.Comment: 14 pages. arXiv admin note: substantial text overlap with arXiv:1607.0407

A remark on the generic vanishing of Koszul cohomology

We give a sufficient condition to study the vanishing of certain Koszul cohomology groups for general pairs $(X,L)\in W^r_{g,d}$ by induction. As an application, we show that to prove the Maximal Rank Conjecture (for quadrics), it suffices to check all cases with the Brill-Noether number $\rho=0$.Comment: 7 pages. Comments are welcom

Comment on "Piezoelectricity in planar boron nitride via a geometric phase"

Using the strain-dependent effective Hamiltonian and the geometric phase, Droth et al. [Phys. Rev. B 94, 075404 (2016)] obtain an analytical expression for the electronic contribution to the piezoelectricity of planar hexagonal boron nitride (h -BN). Their analytical results of piezoelectric constants for h -BN are invalid because of the mistakes in constructing the adiabatic process of the piezoelectricity. In this comment, we reconstruct a proper adiabatic process for piezoelectricity and formulate a general Berry phase expression for the piezoelectric coefficients of two-dimensional piezoelectric crystals by means of the modern theory of polarization. The corrected analytical results of the piezoelectric constants are in complete consistency with the first-principles calculations and hence manifest the validity and generality of the Berry phase expression of piezoelectric coefficients

Layouts for Plane Graphs on Constant Number of Tracks

A \emph{$k$-track} layout of a graph consists of a vertex $k$ colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. A \emph{$k$-queue} layout of a graph consists of a total order of the vertices, and a partition of the edges into $k$ sets such that no two edges that are in the same set are nested with respect to the vertex ordering. The \emph{track number} (\emph{queue number}) of a graph $G$, is the minimum $k$ such that $G$ has a $k$-track ($k$-queue) layout. This paper proves that every $n$-vertex plane graph has constant-bound track and queue numbers. The result implies that every plane has a 3D crossing-free straight-line grid drawing in $O(n)$ volume. The proof utilizes a novel graph partition technique.Comment: arXiv admin note: text overlap with arXiv:1302.0304 by other author