There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Convex bodies: the brunn–minkowski theory

https://www.polyu.edu.hk/ama/staff/new/qilq/qi-eigen-1.pdf

Eigenvalues of a real supersymmetric tensor

第12次 아시아ㆍ太平洋 矯政局長會議 參加記

Linear differential equations form the central topic of this volume, Galois theory being the unifying theme.
A large number of aspects are presented: algebraic theory especially differential Galois theory, formal theory, classification, algorithms to decide solvability in finite terms, monodromy and Hilbert's 21st problem, asymptotics and summability, the inverse problem and linear differential equations in positive characteristic. The appendices aim to help the reader with concepts used, from algebraic geometry, linear algebraic groups, sheaves, and tannakian categories that are used.
This volume will become a standard reference for all mathematicians in this area of mathematics, including graduate students.

Galois theory of linear differential equations

We show that the method of moving quadrics for implicitizing surfaces in ℙ3 applies in certain cases where base points are present. However, if the ideal defined by the parametrization is saturated, then this method rarely applies. Instead, we show that when the base points are a local complete intersection, the implicit equation can be computed as the resultant of the first syzygies.

Implicitization of surfaces in ℙ3 in the presence of base points

Explicit formulas for the multivariate resultant

We present formulas for computing the resultant of sparse polynomials as a quotient of two determinants, the denominator being a minor of the numerator. These formulas extend the original formulation given by Macaulay for homogeneous polynomials.

/pdf/macaulay-style-formulas-for-sparse-resultants-4870a23vtw.pdf

Macaulay style formulas for sparse resultants

We present bounds for the degree and the height of the polynomials arising in some central problems in effective algebraic geometry including the implicitization of rational maps and the effective Nullstellensatz over a variety. Our treatment is based on arithmetic intersection theory in products of projective spaces and extends to the arithmetic setting constructions and results due to Jelonek. A key role is played by the notion of canonical mixed heights of multiprojective varieties. We study this notion from the point of view of resultant theory and establish some of its basic properties, including its behavior with respect to intersections, projections and products. We obtain analogous results for the function field case, including a parametric Nullstellensatz.

/pdf/heights-of-varieties-in-multiprojective-spaces-and-18vfcz22g3.pdf

Heights of varieties in multiprojective spaces and arithmetic Nullstellensatze

Introduction. Inthis paper we make a detailed analysis of the -hypergeometric system (or GKZ system) associated with a monomial curve and integral, hence resonant, exponents. We describe all rational solutions and show in Theorem 1.10 that they are, in fact, Laurent polynomials. We also show that for any exponent there are at most two linearly independent Laurent solutions and that the upper bound is reached if and only if the curve is not arithmetically Cohen-Macaulay. We then construct, for all integral parameters, a basis of local solutions in terms of the roots of the generic univariate polynomial (0.5) associated with . We also determine in Theorem 3.7 the holonomic rank r(α)for all α ∈ Z 2 and show that d ≤ r(α)≤ d +1, where d is the degree of the curve. Moreover, the value d +1 is attained only for those exponents α for which there are two linearly independent rational solutions, and, therefore, r(α)= d for all α if and only if the curve is arithmetically Cohen-Macaulay. Inorder to place these results intheir appropriate con text, we recall the defin ition

/pdf/the-a-hypergeometric-system-associated-with-a-monomial-curve-3ftdhh6qmc.pdf

Carlos D'Andrea

Papers

Implicitization of surfaces in ℙ3 in the presence of base points

Explicit formulas for the multivariate resultant

Macaulay style formulas for sparse resultants

Heights of varieties in multiprojective spaces and arithmetic Nullstellensatze

The A-hypergeometric system associated with a monomial curve