Abstract. A complex frame is a collection of vectors that span C M and define measurements, called intensity measurements, on vectors in C M . In purely mathematical terms, the problem of phase retrieval is to recover a complex vector from its intensity measurements, namely the modulus of its inner product with these frame vectors. We show that any vector is uniquely determined (up to a global phase factor) from 4M − 4 generic measurements. To prove this, we identify the set of frames defining non-injective measurements with the projection of a real variety and bound its dimension.

No abstract

A graded K-algebra R is said to be Koszul if the minimal R-free graded resolution of K is linear. In this paper we study the Koszul property of the homogeneous coordinate ring R of a set of s points in the complex projective space P n . Kempf proved that R is Koszul if s ≤ 2n and the points are in general linear position. If the coordinates of the points are algebraically independent over Q, then we prove that R is Koszul if and only if s ≤ 1 + n + n 2 /4. If s ≤ 2n and the points are in linear general position, then we show that there exists a system of coordinates x 0 , . . . , x n of P n such that all the ideals (x 0 , x 1 , . . . , x i ) with 0 ≤ i ≤ n have a linear R-free resolution.

Let V be closed subscheme of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] defined by a homogeneous ideal I ⊆ A = K [ X 1 , . . . , X n ], and let X be the ( n - 1)-fold obtained by blowing-up [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] along V . If one embeds X in some projective space, one is led to consider the subalgebra K [( I e ) c ] of A for some positive integers c and e . The aim of this paper is to study ring-theoretic properties of K [( I e ) c ]; this is achieved by developing a theory which enables us to describe the local cohomology of certain modules over generalized Segre products of bigraded algebras. These results are applied to the study of the Cohen-Macaulay property of the homogeneous coordinate ring of the blow-up of the projective space along a complete intersection. We also study the Koszul property of diagonal subalgebras of bigraded standard k -algebras.

Abstract. We determine the Hubert function of a determinantal ring and of its canonical module using a combinatorial result of Krattenthaler. This gives a new proof of Abhyankar's formula.Let AT be a field and X = (X¡j) be an m x «-matrix of indeterminates, with m < n . We denote by K[X] the polynomial ring over K in the indeterminates Xij and by Ir+x(X) the ideal in K[X] generated by the r+ 1 -minors of X andThe purpose of this note is to derive a compact formula for the Hubert series of Rr+X and its canonical module. The result is actually a simple rewriting of Abhyankar's formula [1]. However, we want to point out that our formula can as well be obtained from a combinatorial result of Krattenthaler [7] and thus gives a new proof of Abhyankar's formula. Of course, the burden of the proof is hidden in the combinatorial part.Krattenthaler counts the nonintersecting paths with a given number of corners. To be precise consider the set of points V = {(i,j): I < i < m, 1 < j < n}. We define a partial order on V by setting (i, j) < (/', /) if i > i' and j < f . Let P, Q £ V ; a maximal chain C in V with end points P and Q will be called a path from P to Q. A corner of C is an element (i, j) £ C for which (i -1, j) and (i, j -I) belong to C as well. The path in Figure 1 on the next page has two corners.Let Pi, Qi, i = I, ... , r, be points of V. A subset W c V is called an r-tuple of nonintersecting paths from P¡ to Q¡ (i -l,...,r) if W = CiUC2U-uCr where each C, is a path from P¡ to Q¡ and where C¡nCj = 0 if i ^ j. The number of corners c(W) of W is the sum of the number of corners of the C,, i=\,... ,r.

CONCA AND DE NEGRI 600 an M-sequence of monomials. A sequence of monomials m , . . . , m in a

No abstract

Let I be a homogeneous ideal of S = K[x 1 , . . . , x n ] and let < be a term order. We prove that if the initial ideal J = in < (I) is radical then the extremal Betti numbers of S/I and of S/J coincide. In particular, depth(S/I) = depth(S/J) and reg(S/I) = reg(S/J).