Conference Agenda

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Session Overview
Location: vonRoll, Fabrikstr. 8, Foyer
for poster session and reception only
Date: Tuesday, 09/Jul/2019
5:15pm - 7:30pmPP: Welcome reception and poster session
vonRoll, Fabrikstr. 8, Foyer 
 

A constructive algorithm for a positive solution to a system of polynomial inequalities

Nida Obatake

Texas A&M University, United States of America

In this poster we present an algorithm to construct a positive point that solves a system of polynomial inequalities. The procedure uses ideas from polyhedral geometry, specifically normal fans of the Newton polytopes of the polynomials. We give an application of this algorithm to a problem from chemical reaction network theory, an area of mathematics that analyzes the behaviors of chemical processes. A major problem in this area is the stability of equilibria of dynamical systems arising from these networks. Here, we focus on a biological signaling network called the ERK network, a model for dual-site phosphorylation and desphorylation of extracellular signal-regulated kinase. The ERK network is known to be bistable and to exhibit oscillations (Rubinstein, Mattingly, Berezhkovskii and Shvarstman, 2016), but a limiting network of the ERK network- the fully processive dual-site network- is known to have a unique, stable steady state (Conradi and Shiu, 2015). We investigate the emergence of oscillations and instability by analyzing certain subnetworks of ERK. A precursor to oscillations is the existence of a Hopf bifurcation, which are characterized by sign conditions on Hurwitz-matrix determinants (Yang, 2002). Thus, finding a Hopf bifurcation amounts to finding a positive solution to a system of polynomial inequalities. We present a solution using our algorithm.


A generalization of Strassen's Positivstellensatz and its application to large deviation theory

Tobias Fritz

Perimeter Institute for Theoretical Physics, Canada

Strassen's Positivstellensatz is a powerful but little known theorem on preordered commutative semirings satisfying a boundedness condition similar to Archimedeanicity. It characterizes the relaxed preorder induced by all monotone homomorphisms to $mathbb{R}_+$ in terms of a condition involving large powers. I will present a generalization and strengthening of Strassen's result and sketch a variety of applications to probability theory, representation theory, graph theory, and information theory. As a generalization, Strassen's boundedness condition by a polynomial growth condition; as a strengthening, I will show two further equivalent characterizations of the homomorphism-induced preorder.

The application to probability gives results on the asymptotic comparison of one random walk relative to another, which generalizes a (weak form of) Cramér's large deviation theorem. This constitutes the first probabilistic characterization of when the moment-generating function of one random variables dominates that of another, in the context of bounded random variables.


A linear method for positive solutions to polynomial systems

Polly Yu

Univeresity of Wisconsin-Madison, United States of America

We propose a linear method to parametrize the positive solutions of a class of real polynomial systems. These complex-balanced systems arise naturally from chemical reaction networks, where the polynomial system is determined by a directed graph embedded in Euclidean space. Given a positive solution, we define a linear feasibility problem to determine whether the polynomial system can be realized as a complex-balanced system. If yes, we exploit its known toric structure to obtain a monomial parametrization of its positive steady states.


A module theoretic perspective on matroids

Colin William Crowley

University of Wisconsin Madison, United States of America

Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces as modules over an idempotent semifield. All together, this provides bridges between the combinatorics of matroids, the algebra of idempotent modules, and the geometry of tropical linear spaces. The goal of this paper is to strengthen and expand these bridges by systematically developing the idempotent module theory of matroids. Applications include a geometric interpretation of strong matroid maps and the factorization theorem; a generalized notion of strong matroid maps, via an embedding of the category of matroids into a category of module homomorphisms; a monotonicity property for the stable sum and stable intersection of tropical linear spaces; a novel perspective of fundamental transversal matroids; and a tropical analogue of reduced row echelon form.


Catalan-many tropical morphisms to metric trees

Alejandro José Vargas De León

University of Bern, Switzerland

We investigate tree gonality of metric graphs, a tropical version of curve gonality (i.e. the minimum degree of a rational map to P1). Our main contribution is to construct and describe a space Φdtrop of tropical morphisms. As a corollary, we get a constructive method to prove that the gonality of every genus g metric graph Γ is at most ⌈ g / 2 + 1 ⌉. When g is even, we construct all the maps realizing the gonality, and a count of these using certain multiplicity gives the g/2-th Catalan number.


Classification of triples of lattice polytopes with a given mixed volume

Christopher Borger

Otto-von-Guericke Universität Magdeburg, Germany

In this poster we present an overview of our algorithm for the classification of triples of lattice polytopes with a given mixed volume m in dimension 3. It is known that the classification can be reduced to the enumeration of so-called irreducible triples, the number of which is finite for fixed m. Following this algorithm, we were able to enumerate all irreducible triples of normalized mixed volume up to 4 that are inclusion-maximal. On algebro-geometric side this produces a classification of generic trivariate sparse polynomial systems with up to 4 solutions in the complex torus, up to monomial changes of variables. By a recent result of Esterov, this leads to a description of all generic trivariate sparse polynomial systems that are solvable by radicals. This is joint work with Gennadiy Averkov and Ivan Soprunov.


Complexity of variety learning

Oliver Gäfvert

KTH Royal Institute of Technology, Sweden

Extracting structure from a point cloud is a fundamental problem in data analysis, and when this structure is coming from polynomial equations we can use the machinery of algebraic geometry. We study the set of closest real algebraic hypersurfaces to a finite set of points, and investigate methods to approximate the closest hypersurface of a given degree. We investigate the complexity of this problem by considering the Euclidean Distance Degree (EDD) of the variety of point configurations lying on a hypersurface of a given degree and how the complexity depends on the number of points in the configuration.


Embeddability of Markov matrices does not depend only on its principal logarithm

Jordi Roca-Lacostena

Universitat Politècnica de Catalunya, Spain

The embedding problem for Markov substitution matrices consists on determining whether the substituition process described by a given Markov matrix can be explained by homogeneous time-continuous models or not. Several results on the topic seem to point out that the embeddability of a matrix with pairwise different eigenvalues is determined by its principal logarithm. In this poster we build the first example, up to our knowledge, disproving this fact. We will work with 4x4 matrices corresponding to DNA nucleotide-substitution models.

To obtain such example we intersect: i) the polyhedral cone of principal logarithms not fulfilling sthocastic embeddability constrains, ii) the polyhedral cone of a certain family of non-principal logarithms fulfilling sthocastic embeddability constrains and iii) the set of real logarithms of Strand Symmetric Model (SSM) substitution matrices. As a result we get a semialgebraic non-empty open set containing embeddable SSM matrices whose principal logarithm is not involved on its embeddability.

Furthermore, this example can be deformed in such a way that we can obtain infinitely many examples for general Markov matrices. This will be proven in a forthcoming paper jointly with Marta Casanellas and Jesús Fernández-Sánchez.


Gröbner Bases for Toric Staged Tree Models

Lamprini Ananiadi

Otto-von-Guericke Universität Magdeburg, Germany

A toric staged tree model is a discrete statistical model defined as the set of probability distributions in the image of a certain monomial parameterization. In this poster we give an explicit description of the toric ideal associated to any toric staged tree model. Using the theory of toric fiber products we show that toric staged trees can be constructed inductively by gluing smaller components. This allows us to explicitly construct a Gröbner basis whose elements can be obtained combinatorially from the tree graph representation of the model.
This is joint work with Eliana Duarte.


Hermitian Determinantal Surfaces and Three-Dimensional Spectrahedra

Roland Daniel Piontek

TU Dortmund University, Germany

Spectrahedra in real three-space are convex bodies whose algebraic boundary is described by a (projective) surface. Unlike in the plane, the geometric properties of the boundary are not fully understood in general. Smooth determinantal surfaces over the complex numbers are very special and have been classified by Catanese and Beauville. Here, we are interested in the case of real hyperbolic surfaces admitting a hermitian determinantal representation, with applications to spectrahedra. The results are part of a dissertation in progress.


Hyperplane Sections on Real Algebraic Curves

Dimitri Manevich

TU Dortmund, Germany

Given a real algebraic curve, we bound the smallest integer N such that any divisor of degree N is linearly equivalent to a totally real effective divisor. In special cases, this problem can be more vividly interpreted as finding a hyperplane intersecting the given curve in real points only. In the presented poster, we show the main ideas for extending the known bounds for curves with many connected components. Further, we illustrate a connection between the formulated problem and a conjecture about unramified curves in odd dimensional projective spaces. The results are part of a dissertation in progress.


Initial degenerations of Grassmannians

Daniel Joseph Corey

University of Wisconsin - Madison, United States of America

Let Gr_0(d,n) denote the open subvariety of the Grassmannian Gr(d,n) consisting of d-1 dimensional subspaces of P^(n-1) meeting the toric boundary transversely. We prove that Gr_0(3,7) is schoen in the sense that all of its initial degenerations are smooth. We use this to show that the Chow quotient of Gr(3,7) by the maximal torus in GL(7) is the log canonical compactification of the moduli space of 7 lines in P^2 in linear general position. This provides a positive answer to a conjecture of Hacking, Keel, and Tevelev from "Geometry of Chow quotients of Grassmannians."


Maximum Likelihood Estimation for Linear Gaussian Covariance Models with One Sample Point

Jane Ivy Coons

North Carolina State University, United States of America

We will discuss the problem of maximum likelihood estimation using a single sample point for the multivariate Gaussian model in which the covariance matrix is assumed to be a linear combination of two generic symmetric matrices. Solving the score equations in this setting leads to interesting applications of the theory of algebraic curves. In particular, we will focus on counting the number of solutions to the score equations as it depends on the size of the covariance matrix.


Multistationarity in Deficiency-one Power-law Kinetic Systems with Reactant-determined Interactions

Noel Fortun

De La Salle University Manila, Philippines

Multistationarity or the existence of multiple equilibria in a chemical reaction network (CRN) is responsible for the switch-like behavior in the system. Generally, discerning whether a CRN has multistationarity is a difficult task as this entails finding multiple positive solutions to a system of nonlinear differential equations that frequently contains unknown parameters. For deficiency-one mass action networks satisfying certain structural properties, multistationarity can be checked using Feinberg’s Deficiency One Algorithm (DOA). This procedure translates the nonlinear problem of determining the multistationarity of a CRN into a problem that takes the form of linear inequality systems. In this contribution, we extend Feinberg’s DOA to embrace deficiency-one CRNs endowed with rate laws more general than mass action kinetics -- i.e., power-law kinetic systems with reactant-determined interactions (denoted by “PL-RDK”). These are kinetic systems with power-law rate functions whose kinetic order vectors are identical for reactions with the same reactant complex. As illustration, we implement the algorithm to a power-law approximation of a model pre-industrial carbon cycle. The application reveals multistationarity in the pre-industrial state of global carbon cycle.


On new families of stable subgroups of affine Cremona groups, their tame homomorphisms and Non-commutative Cryptography.

Vasyl Alex Ustymenko

University of Maria Curie Sklodowska, Poland

Non-commutative cryptography studies cryptographic primitives and systems which are based on algebraic structures like groups, semigroups and noncommutative rings. It is intensively developing area due due to efforts of G. Maze, G. Monico, J, Rosenthal, P.H. Kropholler, V. Shpilrain, A. Myasnikov, A. Ushakov, D. Kahrobaei, B. Khan, S.Blackburn, S. Galbraith and others. In the case of group the most popular way of group presentation is usage of generation and relations. Classical objects like braid groups, Tompson groups and Grigorchuk groups have been been identified as potential candidates for cryptographic post quantum applications. One of recent directions is the intersection of Non-commutative and Multivariate Cryptographies formed by studies of subgroups (subsemigroups) affine Cremona group (semigroup and semigroup given by their generators in a standard forms of multivariate maps. Some general schemes of key exchange protocols and El Gamal type cryptosystems were recently defined jn terms of stable subgroups and subsemigroups and their homomorphisms. The talk is devoted to new explicit constructions of sequences of stable subsemigroups of affine Cremona semigroup formed by transformation of degree bounded by small constant c (c=2,3) together with their special homomorphisms. Their usage as platforms for cryptographical algorithms will be presented.


Parameter identifiability for ODE models via an input-output representation

Gleb Pogudin

New York University, United States of America

Systems of parametric ordinary differential equations are often used in modeling. One of the challenges in designing models using such systems is structural identifiability, which can be described as follows. The values of some select parameters might be of special interest due to their importance. Usually one tries to determine their numerical values (identify them) by collecting input and output data. However, due to the structure of the model, it can be impossible to determine the parameters from input and output data. When this happens, the parameters are said to be ``not identifiable''. In the process of model design, it is often crucial to know whether the parameters of interest in a potential model are identifiable.

There have been intensive efforts on this challenge since the 1970s made within different communities (e.g. math biology, control theory, symbolic computation). One significant contribution is an approach via an input-output representation of the system proposed in the late 80s. Since then, several algorithms for assessing identifiability based on this approach were designed. DAISY and COMBOS, two modern software packages, are based on such algorithms.

Recent examples by Hong, Ovchinnikov, Pogudin, and Yap show that the underlying assumption of the approach via an input-output representation (called solvability) might be violated even in small linear system. This observation poses two challenges:

  1. Design an algorithm that will determine whether the input-output approach is applicable to a given system.

  2. Prove theorems implying the applicability of the input-output approach to classes of systems of practical importance (e.g. linear compartment models).

In the talk, we will describe our new results in these two directions and discuss remaining open problems.

This is joint work with Alexey Ovchinnikov and Peter Thompson.


Probabilistic analysis on Macaulay matrices over finite fields and complexity of constructing Gröbner bases

Andrea Tenti

University of Bergen, Norway

Problems in cryptanalysis may be reduced to solving a system of multivariate polynomial equations over a finite field $F_q$. Such systems are sometimes overdetermined. E.g., that holds for AES (Advanced Encryption Standard). Gröbner basis methods may be employed to solve the equations, but their complexity is poorly understood. A key parameter is the degree of regularity $d_{reg}$ for the leading forms of the polynomials.

Let $I$ be an ideal in $R=F_q[x_1,…,x_n]/(x_i^q)$ generated by the leading forms of the polynomials. Let $I_d$ be the vector space over $F_q$ generated by the forms in $I$ of degree $d$. The degree of regularity $d_{reg}$ of $I$ is the smallest integer $d$ for which $I_d$ is equal to $R_d$. We prove that time-complexity of constructing a Groebner basis and therefore solving the system is polynomial in the number of monomials of degree ≤d_{reg}. Besides an upper bound on $d_{reg}$ for a sufficiently overdetermined system of polynomials with coefficients in $F_q$ is proved. Their leading forms are of the same degree $D$ are taken uniformly at random. We do not impose any other restrictions. The bound holds with probability tending to $1$ and depends only on $n$, $m$, $D$. Therefore almost all equation systems are solvable in polynomial time if $m$ is large enough compared to $n$. E.g., $m approx n^2/6$ random quadratic equations over $F_2$ have $d_{reg}=3$ with probability close to $1$ and may be solved in time $O(n^{14})$.

Our result complies with the one by Bardet, Faugère, and Salvy of 2004. They computed $d_{reg}$ for a class of systems over $F_2$ called semiregular and conjectured that a random system is semiregular with probability tending to 1. A consequence of our result is that a random system over $F_2$ has the same degree of regularity as a semiregular one. Joint work with Igor Semaev.


The colorful interior of families of convex bodies and its tropical analogue

Marin Boyet

INRIA École Polytechnique, France

Given convex bodies (C1,...,Cn) of Rn that we think as distinct "colors" classes, we say that a vector y in Rn is "rainbow" if every decomposition of y as non-negative linear combination of vectors of C1,...,Cn uses at least one vector in each color class. The "colorful interior" of C1,...,Cn is the set of rainbow vectors.
Using results from convex geometry and topology showed by Cappell, Goodman, Pach, Pollack, Sharir and Wenger, we prove that if the conic hull of all the color classes is pointed, then the colorful interior is either empty or the interior of a simplicial and cone, and coincides with the intersection of all open tranversals of (C1,...,Cn) studied by Lawrence and Soltan.
This is motivated by fixed point problems associated to tropical polynomial maps or to real polynomial maps with nonnegative coefficients: an exact solution of the tropical problem, or an approximate solution of the real version, can be obtained in polynomial time if one knows a point in the colorful interior of an associated family of Newton polytopes.
Finally, we study the tropical anologue of the colorful interior, assuming that (C1,...,Cn) are now tropical polytopes. We show that a tropical rainbow vector exists if all the colored tropical polytopes can be separated from each other by a common tropical hyperplane, and relate this property with the tropical SVM problem, introduced by Gärtner, Jaggi and Tabera.


The Configuration Space and Kinematics of the Canfield Joint

Christian Bueno

University of California, Santa Barbara and NASA Glenn Research Center

The Canfield joint is a novel 3 degrees-of-freedom parallel robotic manipulator is being explored by NASA's Integrated Radio and Optical Communications (iROC) project for deep space communication. As a robotic linkage, we describe its configuration space as a real algebraic set with a natural projection map to the 3-torus. Additionally, we describe a birationally equivalent space that is easier to work with which factors the projection map. Since global sections (right inverses) of this map correspond to the forward kinematics of the Canfield joint, we investigate when the fibers of this projection map can be positive-dimensional. These special points on the 3-torus would correspond to control settings in which the Canfield joint forward kinematics is undefined. These techniques and methodology can be extended naturally to similar non-traditional robotics platforms whose configuration spaces and kinematics are not yet determined.

Separately, we impose a symmetry condition motivated by the practical usage of the Canfield joint and provide algorithms for the solutions of various inverse kinematic problems. In particular, we show that when considering all possible symmetric Canfield joints, the locus of distal centers that would successfully point to an object generically form a nodal cubic curve and otherwise form a sphere union a line.


Limits of Voronoi Decompositions

Madeline Brandt

University of California, Berkeley, United States of America

Voronoi diagrams of finite point sets partition space into regions. Each region contains all points which are nearest to one point in the finite point set. Voronoi diagrams (and their generalizations and variations) have been an object of interest for hundreds of years by mathematicians spanning many fields, and they have numerous applications across the sciences. Recently, Cifuentes, Ranestad, Sturmfels, and Weinstein defined Voronoi cells of varieties, in which the finite point set is replaced by a real algebraic variety. Each point y on the variety has a cell of points in the ambient space corresponding to those points which are closer to y than any other point on the variety. In this poster, we present the limiting behavior of Voronoi diagrams of finite sets, where the finite sets are sampled from the variety and the sample size increases. In this setting, we observe that many interesting features of the variety emerge. This is joint work with Maddie Weinstein.


Multistationarity in the space of total concentrations for systems that admit a monomial parametrization

Alexandru Iosif

OvGU Magdeburg, Germany

We apply tools from real algebraic geometry to the problem of multistationarity of chemical reaction networks. For systems whose steady states admit a monomial parameterization we show that in the space of total concentrations multistationarity is scale invariant. Moreover, for these networks it is possible to decide about multistationarity independent of the rate constants by formulating semialgebraic conditions that involve only total concentrations. Hence quantifier elimination may give new insights into multistationarity regions in the space of total concentrations. To demonstrate this, we show that for the distributive phosphorylation of a protein at two binding sites multistationarity is only possible if the total concentration of the substrate is larger than either the concentration of the kinase or the phosphatase. This result is enabled by the chamber decomposition from polyhedral geometry.

This is joint work with Carsten Conradi and Thomas Kahle (arXiv:1810.08152).


Rhomboid Designs for Linear Regression with Correlated Random Coefficients

Frank Röttger

OVGU Magdeburg, Germany

We study a linear regression model Yi(xi) = f(xi)Tbi on the hypercube with a linear intercept where bi ~ N(β,D) and all Yi(xi) are independent, which means that there is only one observation per realisation of bi. The parameter to be estimated is β, while D is fixed. We assume that the structure of D displays an independent linear intercept and a completely symmetric covariance matrix for the random coefficients. Through a model transformation and the introduction of rhomboid designs, we see that the Kiefer-Wolfowitz equivalence theorem implies when the optimality regions of these designs are either algebraic varieties intersected with trivial constraints or semi-algebraic sets. In fact, it shows that this distinction depends on the choice of the design points. Consequently we then discuss up to dimension 4, for which covariance matrices an optimal rhomboid design is supported either completely on the vertices of the hypercube or has support points in the interior. Furthermore, we conjecture a similar result for arbitrary dimension. This is joint work with Ulrike Graßhoff, Heinz Holling and Rainer Schwabe.


Selecting Minimum Explaining Variables by Pruned Primary Ideal Decomposition with Recursive Calls

Keiji Miura

Kwansei Gakuin University, Japan

Jarrah et al. (2007) proposed an algorithm using the primary decomposition of monomial ideals for selecting minimum wiring diagrams for biological gene networks (or input-output relationships with polynomial functions in general) from finite observations of a set of variables. However, its computational cost with computer algebra system is relatively high, preventing the practical applications to the big data. Here we implemented the algorithm in the form of recursive calls in Matlab and approximated it by pruning the search trees to consider only the cases with the minimum number of explaining variables. This speed-up enabled us to treat larger data of about 100x100 size.


Species Subsets and Embedded Networks of S-systems

Angelyn Relucio Lao

De La Salle University, Philippines

Magombedze and Mulder (2013) studied the gene regulatory system of Mycobacterium Tuberculosis (Mtb) by partitioning this into three subsystems based on putative gene function and role in dormancy/latency development. Each subsystem, in the form of S-system, is represented by an embedded chemical reaction network (CRN), defined by a species subset and a reaction subset induced by the set of digraph vertices of the subsystem. Based on the network decomposition theory initiated by Feinberg in 1987, we have introduced the concept of incidence-independent and developed the theory of C- and C*-decompositions including their structure theorems in terms of linkage classes. With the S-system CRN N of Magombedze and Mulder's Mtb model, its reaction set partition induced decomposition of subnetworks that are not CRNs of S-system but constitute independent decomposition of N. We have also constructed a new S-system CRN N for which the embedded networks are C*-decomposition. We have shown that subnetworks of N and the embedded networks (subnetworks of N*) are digraph homomorphisms. Lastly, we attempted to explore modularity in the context of CRN.


TensorFox

Felipe Bottega Diniz

Universidade Federal do Rio de Janeiro, Brazil

Computing the canonical polyadic decomposition of a tensor is currently a challenging problem. Several approaches have been suggested. Nonlinear least square (NLS) algorithms - more precisely, damped Gauss-Newton (dGN) algorithms - are known to have good convergence properties. Unfortunately they require inverting large Hessian matrices, and for this reason there are just a few implementations of dGN methods. We propose a faster implementation with lower computational and memory costs. Our software was compared against other state of the art implementations.


Topological analysis of neural spike data

Andrea Guidolin

BCAM, Spain

Topological methods have drawn increasing interest in data science, since they allow to extract information from the data and to summarize it in the form of topological invariants that complement the features detected via other techniques. In particular, methods based on the computation of the homology of (a filtration of) suitable objects associated with the data have proved themselves versatile, powerful and computationally treatable. In the field of neuroscience, algebraic topology has been successfully employed for example to detect co-activation patterns of neurons, to investigate functional and structural brain networks and to understand the neural code.

In this poster, we study how algebraic topological techniques can be applied to extract information from spike train data. We consider various possibilities to endow a collection of spike trains with a metric, which can be used to construct a filtration of simplicial complexes in standard ways. Then we compute the invariants provided by algebraic topology and compare them to highlight which type of information they are able to detect. We focus our study on the method of persistent homology and on the computation of Betti curves, showing that statistical properties of the invariants provided by these methods have a strong discriminative power, which can be particularly useful in neural data analysis. Among the number of possible applications in neural spike analysis suggested by our results, we devote particular attention to visual data.


Torus quotient of Richardson varieties in Orthogonal and Symplectic Grassmannians

ARPITA NAYEK

INDIAN INSTITUTE OF TECHNOLOGY, KANPUR, INDIA, India

For any simple, simply connected algebraic group $G$ of type $B,C$ and $D$ and for any maximal parabolic subgroup $P$ of $G$, we provide a criterion for a Richardson variety in $G/P$ to admit semistable points for the action of a maximal torus $T$ with respect to an ample line bundle on $G/P$.


Unboundedness of Markov complexity of monomial curves in A^n for n≥ 4

Dimitra Kosta

University of Glasgow, United Kingdom

Computing the complexity of Markov bases is an extremely challenging problem; no formula is known in general and there are very few classes of toric ideals for which the Markov complexity has been computed. A monomial curve C in A^3 has Markov complexity m(C) two or three. Two if the monomial curve is complete intersection and three otherwise. Our main result shows that there is no d in N such that m(C) ≤ d for all monomial curves C in A^4. The same result is true even if we restrict to complete intersections. We extend this result to all monomial curves in A^n, n ≥ 4.