Boris Shapiro | Stockholm University (original) (raw)

Papers by Boris Shapiro

arXiv (Cornell University), Feb 2, 2017

arXiv (Cornell University), Apr 24, 2022

International Journal of Mathematics, Mar 8, 2022

As was pointed out by S. Karp, Theorem B of paper [V. Sedykh and B. Shapiro, On two conjectures c... more As was pointed out by S. Karp, Theorem B of paper [V. Sedykh and B. Shapiro, On two conjectures concerning convex curves, Int. J. Math. 16(10) (2005) 1157–1173] is wrong. Its claim is based on an erroneous example obtained by multiplication of three concrete totally positive [Formula: see text] upper-triangular matrices, but the order of multiplication of matrices used to produce this example was not the correct one. Below we present a right statement which claims the opposite to that of Theorem B. Its proof can be essentially found in a recent paper [N. Arkani-Hamed, T. Lam and M. Spradlin, Non-perturbative geometries for planar [Formula: see text] SYM amplitudes, J. High Energy Phys. 2021 (2021) 65].

arXiv (Cornell University), Feb 26, 2019

In this paper we settle a special case of the Grassmann convexity conjecture formulated in [13]. ... more In this paper we settle a special case of the Grassmann convexity conjecture formulated in [13]. We present a conjectural formula for the maximal total number of real zeros of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real time (compare with [15]). We show that this formula gives the lower bound for the required total number of real zeros for equations of an arbitrary order and, using our results on the Grassmann convexity, we prove that the aforementioned formula is correct for equations of orders 4 and 5.

arXiv (Cornell University), Dec 28, 2020

As was pointed out by S. Karp, Theorem B of paper [5] is wrong. Its claim is based on an erroneou... more As was pointed out by S. Karp, Theorem B of paper [5] is wrong. Its claim is based on an erroneous example obtained by multiplication of three concrete totally positive 4 × 4 upper-triangular matrices, but the order of multiplication of matrices used to produce this example was not the correct one. Below we present a right statement which claims the opposite to that of Theorem B. Its proof can be essentially found in a recent paper [1].

arXiv (Cornell University), Dec 27, 2016

The space P ol d ≃ CP d of all complex-valued binary forms of degree d (considered up to a consta... more The space P ol d ≃ CP d of all complex-valued binary forms of degree d (considered up to a constant factor) has a standard stratification, each stratum of which contains all forms whose set of multiplicities of their distinct roots is given by a fixed partition µ ⊢ d. For each such stratum Sµ, we introduce its secant degeneracy index ℓµ which is the minimal number of projectively dependent pairwise distinct points on Sµ, i.e., points whose projective span has dimension smaller than ℓµ − 1. In what follows, we discuss the secant degeneracy index ℓµ and the secant degeneracy index ℓμ of the closureSµ.

arXiv (Cornell University), Sep 3, 2004

We discuss the problem of finding an upper bound for the number of equilibrium points of a potent... more We discuss the problem of finding an upper bound for the number of equilibrium points of a potential of several fixed point charges in R n. This question goes back to J. C. Maxwell [10] and M. Morse [12]. Using fewnomial theory we show that for a given number of charges there exists an upper bound independent of the dimension, and show it to be at most 12 for three charges. We conjecture an exact upper bound for a given configuration of nonnegative charges in terms of its Voronoi diagram, and prove it asymptotically.

arXiv (Cornell University), Feb 21, 2022

In this paper, we initiate the study of a new interrelation between linear ordinary differential ... more In this paper, we initiate the study of a new interrelation between linear ordinary differential operators and complex dynamics which we discuss in details in the simplest case of operators of order 1. Namely, assuming that such an operator T has polynomial coefficients, we interpret it as a continuous family of Hutchinson operators acting on the space of positive powers of linear forms. Using this interpretation of T , we introduce its continuously Hutchinson invariant subsets of the complex plane and investigate a variety of their properties. In particular, we prove that for any T with non-constant coefficients, there exists a unique minimal under inclusion invariant set M T CH. Further, we completely characterize the class of operators T for which M T CH is compact and find it explicitly for several special types of operators. In particular, we present strong evidence that the boundary of M T CH is piecewise analytic in contrast to the boundaries of classical invariant sets occurring in complex dynamics.

Experimental Mathematics, Dec 1, 2012

Motivated by the necessities of the invariant theory of binary forms J. J. Sylvester constructed ... more Motivated by the necessities of the invariant theory of binary forms J. J. Sylvester constructed in 1878 for each graph with possible multiple edges but without loops its symmetrized graph monomial which is a polynomial in the vertex labels of the original graph. In the 20-th century this construction was studied by several authors. We pose the question for which graphs this polynomial is a non-negative resp. a sum of squares. This problem is motivated by a recent conjecture of F. Sottile and E. Mukhin on discriminant of the derivative of a univariate polynomial, and an interesting example of P. and A. Lax of a graph with 4 edges whose symmetrized graph monomial is non-negative but not a sum of squares. We present detailed information about symmetrized graph monomials for graphs with four and six edges, obtained by computer calculations. 2. Some generalities about symmetrized graph monomials Definition 2. Let g be a directed graph with d edges and vertices v 1 , v 2 ,. .. , v n. Let α = (α 1 ,. .. , α d) be an integer partition of d. A partition-coloring of g with α is an assignment of colors to the edges and vertices of g satisfying the following: • For each color i, 1 ≤ i ≤ d, we paint the vertex v j and α i edges connected to v j with the color i. • Each edge of g is painted with exacly one color. • Each vertex is painted at most once. An edge is odd-colored if it has color j and is directed to a vertex with the same color. The coloring is said to be negative if there is an odd number of odd-colored edges in g, and positive otherwise. Definition 3. Given a polynomial P (x) and a multi-index α = (α 1 ,. .. , α n), we use the notation Coef f α (P (x)) to denote the coefficient in front of x α in P (x). Note that we may view α as a partition of the sum of the indices. Lemma 4. Let g be a directed graph with d edges and vertices v 1 , v 2 ,. .. , v n. Then Coef f α (g) is given by the number of positive partition-colorings of g with α minus the number of negative partition-colorings. Proof. See [10, Lemma 2.3].

Linear Algebra and its Applications, Apr 1, 2014

Around multivariate Schmidt-Spitzer theorem.

HAL (Le Centre pour la Communication Scientifique Directe), 2011

The well-known Hermite-Biehler theorem claims that a univariate monic polynomial s of degree k ha... more The well-known Hermite-Biehler theorem claims that a univariate monic polynomial s of degree k has all roots in the open upper half-plane if and only if s = p + iq where p and q are real polynomials of degree k and k − 1 resp. with all real, simple and interlacing roots, and q has a negative leading coefficient. Considering roots of p as cyclically ordered on RP 1 we show that the open disk in CP 1 having a pair of consecutive roots of p as its diameter is the maximal univalent disk for the function R = q p. This solves a special case of the so-called Hermite-Biehler problem.

Arnold mathematical journal, Jul 31, 2018

In what follows, we present a large number of questions which were posed on the problem solving s... more In what follows, we present a large number of questions which were posed on the problem solving seminar in algebra at Stockholm University during the period Fall 2014-Spring 2017 along with a number of results related to these problems. Many of the results were obtained by participants of the latter seminar. Keywords Waring problem for forms • Generic and maximal ranks • Ideals of generic forms • Power ideals • Lefschetz properties • Symbolic powers 1 The Waring Problem for Complex-Valued Forms The following famous result on binary forms was proven by Sylvester in 1851. Below we use the terms "forms" and "homogeneous polynomials" as synonyms.

arXiv (Cornell University), Mar 13, 2015

In this note we show that a linear ordinary differential equation with polynomial coefficients is... more In this note we show that a linear ordinary differential equation with polynomial coefficients is globally non-oscillating in CP 1 if and only if it is Fuchsian, and at every its singular point any two distinct characteristic exponents have distinct real parts. As a byproduct of our study, we obtain a new explicit upper bound for the number of zeros of exponential polynomials in a horizontal strip.

Experimental Mathematics, 2001

where Q(x) is some fixed polynomial of degree k. One can eas-Acknowledgement j|y see that ^ hag e... more where Q(x) is some fixed polynomial of degree k. One can eas-Acknowledgement j|y see that ^ hag exact)y one p O | ynom i a | eigenfunction p n (x) References jn eacn ^eg ree n > 0 and its eigenvalue A n/ k equals (n + k)!/n!. A more intriguing fact is that all zeros of p n (x) lie in the convex hull of the set of zeros to Q(x). In particular, if Q(x) has only real zeros then each p n (x) enjoys the same property. We formulate a number of conjectures on different properties of p n (x) based on computer experiments as, for example, the interlacing property, a formula for the asymptotic distribution of zeros etc. These polynomial eigenfunctions might be thought of as a generalization of the classical Gegenbauer polynomials with half-integer superscript, this case arising when our Q(x) is an integer power ofx 2-1.

Journal D Analyse Mathematique, 2011

We prove a parametric generalization of the classical Poincaré-Perron theorem on stabilizing recu... more We prove a parametric generalization of the classical Poincaré-Perron theorem on stabilizing recurrence relations where we assume that the varying coefficients of a recurrence depend on auxiliary parameters and converge uniformly in these parameters to their limiting values. As an application we study convergence of the ratios of families of functions satisfying finite recurrence relations with varying functional coefficients. For example, we explicitly describe the asymptotic ratio for two classes of biorthogonal polynomials introduced by Ismail and Masson.

Proceedings of The London Mathematical Society, Jun 1, 2007

We discuss the problem of finding an upper bound for the number of equilibrium points of a potent... more We discuss the problem of finding an upper bound for the number of equilibrium points of a potential of several fixed point charges in R n. This question goes back to J. C. Maxwell [10] and M. Morse [12]. Using fewnomial theory we show that for a given number of charges there exists an upper bound independent of the dimension, and show it to be at most 12 for three charges. We conjecture an exact upper bound for a given configuration of nonnegative charges in terms of its Voronoi diagram, and prove it asymptotically.

Journal of Dynamical and Control Systems, Dec 24, 2021

As was pointed out by S. Karp, Theorem B of paper [5] is wrong. Its claim is based on an erroneou... more As was pointed out by S. Karp, Theorem B of paper [5] is wrong. Its claim is based on an erroneous example obtained by multiplication of three concrete totally positive 4 × 4 upper-triangular matrices, but the order of multiplication of matrices used to produce this example was not the correct one. Below we present a right statement which claims the opposite to that of Theorem B. Its proof can be essentially found in a recent paper [1].

arXiv (Cornell University), Apr 23, 2016

For a point p ∈ CP 2 and a triple (g, d, ℓ) of non-negative integers we define a Hurwitz-Severi n... more For a point p ∈ CP 2 and a triple (g, d, ℓ) of non-negative integers we define a Hurwitz-Severi number H g,d,ℓ as the number of generic irreducible plane curves of genus g and degree d + ℓ having an ℓ-fold node at p and at most ordinary nodes as singularities at the other points, such that the projection of the curve from p has a prescribed set of local and remote tangents and lines passing through nodes. In the cases d + ℓ ≥ g + 2 and d + 2ℓ ≥ g + 2 > d+ℓ we express the Hurwitz-Severi numbers via appropriate ordinary Hurwitz numbers. The remaining case d + 2ℓ < g + 2 is still widely open.

arXiv (Cornell University), Sep 7, 2009

We prove a parametric generalization of the classical Poincaré-Perron theorem on stabilizing recu... more We prove a parametric generalization of the classical Poincaré-Perron theorem on stabilizing recurrence relations where we assume that the varying coefficients of a recurrence depend on auxiliary parameters and converge uniformly in these parameters to their limiting values. As an application we study convergence of the ratios of families of functions satisfying finite recurrence relations with varying functional coefficients. For example, we explicitly describe the asymptotic ratio for two classes of biorthogonal polynomials introduced by Ismail and Masson.

arXiv (Cornell University), Dec 7, 2005

Generalizing the classical matrix-tree theorem we provide a formula counting subgraphs of a given... more Generalizing the classical matrix-tree theorem we provide a formula counting subgraphs of a given graph with a fixed 2-core. We use this generalization to obtain an analog of the matrix-tree theorem for the root system Dn (the classical theorem corresponds to the An-case). Several byproducts of the developed technique, such as a new formula for a specialization of the multivariate Tutte polynomial, are of independent interest.

arXiv (Cornell University), Feb 2, 2017

arXiv (Cornell University), Apr 24, 2022

International Journal of Mathematics, Mar 8, 2022

As was pointed out by S. Karp, Theorem B of paper [V. Sedykh and B. Shapiro, On two conjectures c... more As was pointed out by S. Karp, Theorem B of paper [V. Sedykh and B. Shapiro, On two conjectures concerning convex curves, Int. J. Math. 16(10) (2005) 1157–1173] is wrong. Its claim is based on an erroneous example obtained by multiplication of three concrete totally positive [Formula: see text] upper-triangular matrices, but the order of multiplication of matrices used to produce this example was not the correct one. Below we present a right statement which claims the opposite to that of Theorem B. Its proof can be essentially found in a recent paper [N. Arkani-Hamed, T. Lam and M. Spradlin, Non-perturbative geometries for planar [Formula: see text] SYM amplitudes, J. High Energy Phys. 2021 (2021) 65].

arXiv (Cornell University), Feb 26, 2019

In this paper we settle a special case of the Grassmann convexity conjecture formulated in [13]. ... more In this paper we settle a special case of the Grassmann convexity conjecture formulated in [13]. We present a conjectural formula for the maximal total number of real zeros of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real time (compare with [15]). We show that this formula gives the lower bound for the required total number of real zeros for equations of an arbitrary order and, using our results on the Grassmann convexity, we prove that the aforementioned formula is correct for equations of orders 4 and 5.

arXiv (Cornell University), Dec 28, 2020

As was pointed out by S. Karp, Theorem B of paper [5] is wrong. Its claim is based on an erroneou... more As was pointed out by S. Karp, Theorem B of paper [5] is wrong. Its claim is based on an erroneous example obtained by multiplication of three concrete totally positive 4 × 4 upper-triangular matrices, but the order of multiplication of matrices used to produce this example was not the correct one. Below we present a right statement which claims the opposite to that of Theorem B. Its proof can be essentially found in a recent paper [1].

arXiv (Cornell University), Dec 27, 2016

The space P ol d ≃ CP d of all complex-valued binary forms of degree d (considered up to a consta... more The space P ol d ≃ CP d of all complex-valued binary forms of degree d (considered up to a constant factor) has a standard stratification, each stratum of which contains all forms whose set of multiplicities of their distinct roots is given by a fixed partition µ ⊢ d. For each such stratum Sµ, we introduce its secant degeneracy index ℓµ which is the minimal number of projectively dependent pairwise distinct points on Sµ, i.e., points whose projective span has dimension smaller than ℓµ − 1. In what follows, we discuss the secant degeneracy index ℓµ and the secant degeneracy index ℓμ of the closureSµ.

arXiv (Cornell University), Sep 3, 2004

We discuss the problem of finding an upper bound for the number of equilibrium points of a potent... more We discuss the problem of finding an upper bound for the number of equilibrium points of a potential of several fixed point charges in R n. This question goes back to J. C. Maxwell [10] and M. Morse [12]. Using fewnomial theory we show that for a given number of charges there exists an upper bound independent of the dimension, and show it to be at most 12 for three charges. We conjecture an exact upper bound for a given configuration of nonnegative charges in terms of its Voronoi diagram, and prove it asymptotically.

arXiv (Cornell University), Feb 21, 2022

In this paper, we initiate the study of a new interrelation between linear ordinary differential ... more In this paper, we initiate the study of a new interrelation between linear ordinary differential operators and complex dynamics which we discuss in details in the simplest case of operators of order 1. Namely, assuming that such an operator T has polynomial coefficients, we interpret it as a continuous family of Hutchinson operators acting on the space of positive powers of linear forms. Using this interpretation of T , we introduce its continuously Hutchinson invariant subsets of the complex plane and investigate a variety of their properties. In particular, we prove that for any T with non-constant coefficients, there exists a unique minimal under inclusion invariant set M T CH. Further, we completely characterize the class of operators T for which M T CH is compact and find it explicitly for several special types of operators. In particular, we present strong evidence that the boundary of M T CH is piecewise analytic in contrast to the boundaries of classical invariant sets occurring in complex dynamics.

Experimental Mathematics, Dec 1, 2012

Motivated by the necessities of the invariant theory of binary forms J. J. Sylvester constructed ... more Motivated by the necessities of the invariant theory of binary forms J. J. Sylvester constructed in 1878 for each graph with possible multiple edges but without loops its symmetrized graph monomial which is a polynomial in the vertex labels of the original graph. In the 20-th century this construction was studied by several authors. We pose the question for which graphs this polynomial is a non-negative resp. a sum of squares. This problem is motivated by a recent conjecture of F. Sottile and E. Mukhin on discriminant of the derivative of a univariate polynomial, and an interesting example of P. and A. Lax of a graph with 4 edges whose symmetrized graph monomial is non-negative but not a sum of squares. We present detailed information about symmetrized graph monomials for graphs with four and six edges, obtained by computer calculations. 2. Some generalities about symmetrized graph monomials Definition 2. Let g be a directed graph with d edges and vertices v 1 , v 2 ,. .. , v n. Let α = (α 1 ,. .. , α d) be an integer partition of d. A partition-coloring of g with α is an assignment of colors to the edges and vertices of g satisfying the following: • For each color i, 1 ≤ i ≤ d, we paint the vertex v j and α i edges connected to v j with the color i. • Each edge of g is painted with exacly one color. • Each vertex is painted at most once. An edge is odd-colored if it has color j and is directed to a vertex with the same color. The coloring is said to be negative if there is an odd number of odd-colored edges in g, and positive otherwise. Definition 3. Given a polynomial P (x) and a multi-index α = (α 1 ,. .. , α n), we use the notation Coef f α (P (x)) to denote the coefficient in front of x α in P (x). Note that we may view α as a partition of the sum of the indices. Lemma 4. Let g be a directed graph with d edges and vertices v 1 , v 2 ,. .. , v n. Then Coef f α (g) is given by the number of positive partition-colorings of g with α minus the number of negative partition-colorings. Proof. See [10, Lemma 2.3].

Linear Algebra and its Applications, Apr 1, 2014

Around multivariate Schmidt-Spitzer theorem.

HAL (Le Centre pour la Communication Scientifique Directe), 2011

The well-known Hermite-Biehler theorem claims that a univariate monic polynomial s of degree k ha... more The well-known Hermite-Biehler theorem claims that a univariate monic polynomial s of degree k has all roots in the open upper half-plane if and only if s = p + iq where p and q are real polynomials of degree k and k − 1 resp. with all real, simple and interlacing roots, and q has a negative leading coefficient. Considering roots of p as cyclically ordered on RP 1 we show that the open disk in CP 1 having a pair of consecutive roots of p as its diameter is the maximal univalent disk for the function R = q p. This solves a special case of the so-called Hermite-Biehler problem.

Arnold mathematical journal, Jul 31, 2018

In what follows, we present a large number of questions which were posed on the problem solving s... more In what follows, we present a large number of questions which were posed on the problem solving seminar in algebra at Stockholm University during the period Fall 2014-Spring 2017 along with a number of results related to these problems. Many of the results were obtained by participants of the latter seminar. Keywords Waring problem for forms • Generic and maximal ranks • Ideals of generic forms • Power ideals • Lefschetz properties • Symbolic powers 1 The Waring Problem for Complex-Valued Forms The following famous result on binary forms was proven by Sylvester in 1851. Below we use the terms "forms" and "homogeneous polynomials" as synonyms.

arXiv (Cornell University), Mar 13, 2015

In this note we show that a linear ordinary differential equation with polynomial coefficients is... more In this note we show that a linear ordinary differential equation with polynomial coefficients is globally non-oscillating in CP 1 if and only if it is Fuchsian, and at every its singular point any two distinct characteristic exponents have distinct real parts. As a byproduct of our study, we obtain a new explicit upper bound for the number of zeros of exponential polynomials in a horizontal strip.

Experimental Mathematics, 2001

where Q(x) is some fixed polynomial of degree k. One can eas-Acknowledgement j|y see that ^ hag e... more where Q(x) is some fixed polynomial of degree k. One can eas-Acknowledgement j|y see that ^ hag exact)y one p O | ynom i a | eigenfunction p n (x) References jn eacn ^eg ree n > 0 and its eigenvalue A n/ k equals (n + k)!/n!. A more intriguing fact is that all zeros of p n (x) lie in the convex hull of the set of zeros to Q(x). In particular, if Q(x) has only real zeros then each p n (x) enjoys the same property. We formulate a number of conjectures on different properties of p n (x) based on computer experiments as, for example, the interlacing property, a formula for the asymptotic distribution of zeros etc. These polynomial eigenfunctions might be thought of as a generalization of the classical Gegenbauer polynomials with half-integer superscript, this case arising when our Q(x) is an integer power ofx 2-1.

Journal D Analyse Mathematique, 2011

We prove a parametric generalization of the classical Poincaré-Perron theorem on stabilizing recu... more We prove a parametric generalization of the classical Poincaré-Perron theorem on stabilizing recurrence relations where we assume that the varying coefficients of a recurrence depend on auxiliary parameters and converge uniformly in these parameters to their limiting values. As an application we study convergence of the ratios of families of functions satisfying finite recurrence relations with varying functional coefficients. For example, we explicitly describe the asymptotic ratio for two classes of biorthogonal polynomials introduced by Ismail and Masson.

Proceedings of The London Mathematical Society, Jun 1, 2007

We discuss the problem of finding an upper bound for the number of equilibrium points of a potent... more We discuss the problem of finding an upper bound for the number of equilibrium points of a potential of several fixed point charges in R n. This question goes back to J. C. Maxwell [10] and M. Morse [12]. Using fewnomial theory we show that for a given number of charges there exists an upper bound independent of the dimension, and show it to be at most 12 for three charges. We conjecture an exact upper bound for a given configuration of nonnegative charges in terms of its Voronoi diagram, and prove it asymptotically.

Journal of Dynamical and Control Systems, Dec 24, 2021

As was pointed out by S. Karp, Theorem B of paper [5] is wrong. Its claim is based on an erroneou... more As was pointed out by S. Karp, Theorem B of paper [5] is wrong. Its claim is based on an erroneous example obtained by multiplication of three concrete totally positive 4 × 4 upper-triangular matrices, but the order of multiplication of matrices used to produce this example was not the correct one. Below we present a right statement which claims the opposite to that of Theorem B. Its proof can be essentially found in a recent paper [1].

arXiv (Cornell University), Apr 23, 2016

For a point p ∈ CP 2 and a triple (g, d, ℓ) of non-negative integers we define a Hurwitz-Severi n... more For a point p ∈ CP 2 and a triple (g, d, ℓ) of non-negative integers we define a Hurwitz-Severi number H g,d,ℓ as the number of generic irreducible plane curves of genus g and degree d + ℓ having an ℓ-fold node at p and at most ordinary nodes as singularities at the other points, such that the projection of the curve from p has a prescribed set of local and remote tangents and lines passing through nodes. In the cases d + ℓ ≥ g + 2 and d + 2ℓ ≥ g + 2 > d+ℓ we express the Hurwitz-Severi numbers via appropriate ordinary Hurwitz numbers. The remaining case d + 2ℓ < g + 2 is still widely open.

arXiv (Cornell University), Sep 7, 2009

We prove a parametric generalization of the classical Poincaré-Perron theorem on stabilizing recu... more We prove a parametric generalization of the classical Poincaré-Perron theorem on stabilizing recurrence relations where we assume that the varying coefficients of a recurrence depend on auxiliary parameters and converge uniformly in these parameters to their limiting values. As an application we study convergence of the ratios of families of functions satisfying finite recurrence relations with varying functional coefficients. For example, we explicitly describe the asymptotic ratio for two classes of biorthogonal polynomials introduced by Ismail and Masson.

arXiv (Cornell University), Dec 7, 2005

Generalizing the classical matrix-tree theorem we provide a formula counting subgraphs of a given... more Generalizing the classical matrix-tree theorem we provide a formula counting subgraphs of a given graph with a fixed 2-core. We use this generalization to obtain an analog of the matrix-tree theorem for the root system Dn (the classical theorem corresponds to the An-case). Several byproducts of the developed technique, such as a new formula for a specialization of the multivariate Tutte polynomial, are of independent interest.

We consider level crossing in a matrix family H = H 0 + λV where H 0 is a fixed N × N matrix and ... more We consider level crossing in a matrix family H = H 0 + λV where H 0 is a fixed N × N matrix and V belongs to one of the standard Gaussian random matrix ensembles. We study the probability distribution of level crossing points in the complex plane of λ, for which we obtain a number of exact, asymptotic and approximate formulas.

For a point p ∈ CP 2 and a triple (g, d,) of non-negative integers, we define a Hurwitz–Severi nu... more For a point p ∈ CP 2 and a triple (g, d,) of non-negative integers, we define a Hurwitz–Severi number H g,d,, as the number of generic irreducible plane curves of genus g and degree d + having an-fold node at p and at most ordinary nodes as singularities at the other points, such that the projection of the curve from p has a prescribed set of local and remote tangents and lines passing through nodes. In the cases d + ≥ g + 2 and d + 2 ≥ g + 2 > d+, we express the Hurwitz–Severi numbers via appropriate ordinary Hurwitz numbers. The remaining case d + 2 < g + 2 is still widely open.

In this paper we study a filtered " K-theoretical " analog of a graded algebra associated to any ... more In this paper we study a filtered " K-theoretical " analog of a graded algebra associated to any loopless graph G which was introduced in [4]. We show that two such filtered algebras are isomorphic if and only if their graphs are isomorphic. We also study a large family of filtered generalizations of the latter graded algebra which includes the above " K-theoretical " analog.