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Trees, parking functions, syzygies, and deformations of monomial ideals

- A. Postnikov, B. Shapiro
- Mathematics
- 11 January 2003

For a graph G, we construct two algebras whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial… Expand

Algebras of Curvature Forms on Homogeneous Manifolds

- A. Postnikov, B. Shapiro, M. Shapiro
- Mathematics
- 19 January 1999

Let C(X) be the algebra generated by the curvature two-forms of standard holomorphic hermitian line bundles over the complex homogeneous manifold X = G=B. The cohomology ring of X is a quotient of… Expand

Mystery of point charges

- A. Gabrielov, D. Novikov, B. Shapiro
- Mathematics, Physics
- 3 September 2004

We discuss the problem of finding an upper bound for the number of equilibrium points of a potential of several fixed point charges in Rn. This question goes back to J. C. Maxwell [10] and M. Morse… Expand

ON THE NUMBER OF CONNECTED COMPONENTS IN THE SPACE OF CLOSED NONDEGENERATE CURVES ON S n

- B. Shapiro, M. Shapiro
- Mathematics
- 1 July 1991

The main definition. A parametrized curve γ : I → R is called nondegenerate if for any t ∈ I the vectors γ′(t), . . . , γ(t) are linearly independent. Analogously γ : I → S is called nondegenerate if… Expand

On ring generated by Chern 2-forms on %plane1D;54A;%plane1D;543;n/B

- B. Shapiro, M. Shapiro
- Mathematics
- 1998

Abstract In this short Note we give an explicit presentation of the ring A n generated by the curvature 2-forms of the standard Hermitian linear bundles over %plane1D;54A;%plane1D;543; n / B as the… Expand

Homotopy classification of nondegenerate quasiperiodic curves on the 2-sphere

- B. Shapiro, B. Khesin
- Mathematics
- 1999

We classify the curves on S with fixed monodromy operator and nowhere vanishing geodesic curvature. The number of connected components of the space of such curves turns out to be 2 or 3 depending on… Expand

On the Waring problem for polynomial rings

- R. Fröberg, G. Ottaviani, B. Shapiro
- Mathematics, Medicine
- Proceedings of the National Academy of Sciences
- 6 December 2011

In this note we discuss an analog of the classical Waring problem for . Namely, we show that a general homogeneous polynomial of degree divisible by k≥2 can be represented as a sum of at most kn k-th… Expand

On Spectral Polynomials of the Heun Equation. II

- B. Shapiro, K. Takemura, M. Tater
- Mathematics
- 12 December 2008

AbstractThe well-known Heun equation has the form
$$\begin{array}{ll}\left\{Q(z)\frac {d^2}{dz^2}+P(z)\frac{d}{dz}+V(z)\right\}S(z)=0,\end{array}$$where Q(z) is a cubic complex polynomial, P(z) and… Expand

Ramified coverings of S^2 with one degenerate branching point and enumeration of edge-ordered graphs

- B. Shapiro, M. Shapiro, A. Vainshtein
- Mathematics
- 1997

Ramified coverings of S^2 with one degenerate branching point and enumeration of edge-ordered graphs

Root Asymptotics of Spectral Polynomials for the Lamé Operator

- J. Borcea, B. Shapiro
- Mathematics
- 30 January 2007

The study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of double-periodic solutions of its Weierstrass form has a long history. Such solutions appear… Expand

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