Andrew J. Sommese
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Andrew J. Sommese has a total of 45 co-authors. Here are the top co-authors by number of shared publications:
- Charles W. Wampler
University of Notre Dame
16 shared publications - Jonathan D. Hauenstein
University of Notre Dame
14 shared publications - Wenrui Hao
Department of Mathematics Penn State University University Park Pennsylvania USA
7 shared publications - Chris Peterson
Department of Mathematics, Colorado State University, Fort Collins, USA
6 shared publications - Daniel J. Bates
Colorado State University
6 shared publications - Jan Verschelde
Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago 851 South Morgan (M/C 249) Chicago 60607-7045 IL USA
6 shared publications
Latest Publications
- 4 August 2025
- Published by Springer Nature in Archiv der Mathematik
- Vol. 125 (4) , 389-397
On a special class of rank two vector bundles on projective surfaces
- 4 August 2025
- Published by Springer Nature in Archiv der Mathematik
- Vol. 125 (4) , 389-397
On a special class of rank two vector bundles on projective surfaces
- Open Access
Number of Real Critical Points of Cyclotomic Polynomials
We study the number of real critical points of a cyclotomic polynomial $\Phi_{n}(x)$, that is, the real roots of $\Phi_{n}^{\prime}(x)$. As usual, one can, without losing generality, restrict $n$ to be the product of distinct odd primes, say $p_{1}<\cdots<p_{k}$. We show that if the primes are "sufficiently separated" then there are exactly $2^{k}-1$ real roots of $\Phi_{n}^{\prime}(x)$ and each of them is simple.Open AccessNumber of Real Critical Points of Cyclotomic Polynomials
We study the number of real critical points of a cyclotomic polynomial $\Phi_{n}(x)$, that is, the real roots of $\Phi_{n}^{\prime}(x)$. As usual, one can, without losing generality, restrict $n$ to be the product of distinct odd primes, say $p_{1}<\cdots<p_{k}$. We show that if the primes are "sufficiently separated" then there are exactly $2^{k}-1$ real roots of $\Phi_{n}^{\prime}(x)$ and each of them is simple. - Open Access
Singular value decomposition of complexes
Singular value decompositions of matrices are widely used in numerical linear algebra with many applications. In this paper, we extend the notion of singular value decompositions to finite complexes of real vector spaces. We provide two methods to compute them and present several applications.
Open AccessSingular value decomposition of complexes
Singular value decompositions of matrices are widely used in numerical linear algebra with many applications. In this paper, we extend the notion of singular value decompositions to finite complexes of real vector spaces. We provide two methods to compute them and present several applications.
- 21 December 2017
- Published by Springer Nature
- p. 107-120
Homotopies for Connected Components of Algebraic Sets with Application to Computing Critical Sets
- 21 December 2017
- Published by Springer Nature
- p. 107-120
Homotopies for Connected Components of Algebraic Sets with Application to Computing Critical Sets
- Open Access
Homotopy techniques for tensor decomposition and perfect identifiability
Let T be a general complex tensor of format $(n_1,...,n_d)$. When the fraction $\prod_in_i/[1+\sum_i(n_i-1)]$ is an integer, and a natural inequality (called balancedness) is satisfied, it is expected that T has finitely many minimal decomposition as a sum of decomposable tensors. We show how homotopy techniques allow us to find all the decompositions of T, starting from a given one. Computationally, this gives a guess regarding the total number of such decompositions. This guess matches exactly with all cases previously known, and predicts several unknown cases. Some surprising experiments yielded two new cases of generic identifiability: formats (3,4,5) and (2,2,2,3) which have a unique decomposition as the sum of 6 and 4 decomposable tensors, respectively. We conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds, i.e., the only identifiable cases. Building on the computational experiments, we use algebraic geometry to prove these two new cases are indeed generically identifiable.Open AccessHomotopy techniques for tensor decomposition and perfect identifiability
Let T be a general complex tensor of format $(n_1,...,n_d)$. When the fraction $\prod_in_i/[1+\sum_i(n_i-1)]$ is an integer, and a natural inequality (called balancedness) is satisfied, it is expected that T has finitely many minimal decomposition as a sum of decomposable tensors. We show how homotopy techniques allow us to find all the decompositions of T, starting from a given one. Computationally, this gives a guess regarding the total number of such decompositions. This guess matches exactly with all cases previously known, and predicts several unknown cases. Some surprising experiments yielded two new cases of generic identifiability: formats (3,4,5) and (2,2,2,3) which have a unique decomposition as the sum of 6 and 4 decomposable tensors, respectively. We conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds, i.e., the only identifiable cases. Building on the computational experiments, we use algebraic geometry to prove these two new cases are indeed generically identifiable. - Open Access
- 18 November 2015
- Published by IOP Publishing in Journal of Physics A: Mathematical and Theoretical
- Vol. 48 (49) , 494003
Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity
We consider the -invariant open spin-1/2 XXZ quantum spin chain of finite length N . For the case that q is a root of unity, we propose a formula for the number of admissible solutions of the Bethe ansatz equations in terms of dimensions of irreducible representations of the Temperley–Lieb algebra; and a formula for the degeneracies of the transfer matrix eigenvalues in terms of dimensions of tilting -modules. These formulas include corrections that appear if two or more tilting modules are spectrum-degenerate. For the XX case ( ), we give explicit formulas for the number of admissible solutions and degeneracies. We also consider the cases of generic q and the isotropic ( ) limit. Numerical solutions of the Bethe equations up to N = 8 are presented. Our results are consistent with the Bethe ansatz solution being complete.
- 18 November 2015
- Published by IOP Publishing in Journal of Physics A: Mathematical and Theoretical
- Vol. 48 (49) , 494003
Open AccessCounting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity
We consider the -invariant open spin-1/2 XXZ quantum spin chain of finite length N . For the case that q is a root of unity, we propose a formula for the number of admissible solutions of the Bethe ansatz equations in terms of dimensions of irreducible representations of the Temperley–Lieb algebra; and a formula for the degeneracies of the transfer matrix eigenvalues in terms of dimensions of tilting -modules. These formulas include corrections that appear if two or more tilting modules are spectrum-degenerate. For the XX case ( ), we give explicit formulas for the number of admissible solutions and degeneracies. We also consider the cases of generic q and the isotropic ( ) limit. Numerical solutions of the Bethe equations up to N = 8 are presented. Our results are consistent with the Bethe ansatz solution being complete.
- Open Access
Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity
We consider the sl(2)_q-invariant open spin-1/2 XXZ quantum spin chain of finite length N. For the case that q is a root of unity, we propose a formula for the number of admissible solutions of the Bethe ansatz equations in terms of dimensions of irreducible representations of the Temperley-Lieb algebra; and a formula for the degeneracies of the transfer matrix eigenvalues in terms of dimensions of tilting sl(2)_q-modules. These formulas include corrections that appear if two or more tilting modules are spectrum-degenerate. For the XX case (q=exp(i pi/2)), we give explicit formulas for the number of admissible solutions and degeneracies. We also consider the cases of generic q and the isotropic (q->1) limit. Numerical solutions of the Bethe equations up to N=8 are presented. Our results are consistent with the Bethe ansatz solution being complete.
Open AccessCounting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity
We consider the sl(2)_q-invariant open spin-1/2 XXZ quantum spin chain of finite length N. For the case that q is a root of unity, we propose a formula for the number of admissible solutions of the Bethe ansatz equations in terms of dimensions of irreducible representations of the Temperley-Lieb algebra; and a formula for the degeneracies of the transfer matrix eigenvalues in terms of dimensions of tilting sl(2)_q-modules. These formulas include corrections that appear if two or more tilting modules are spectrum-degenerate. For the XX case (q=exp(i pi/2)), we give explicit formulas for the number of admissible solutions and degeneracies. We also consider the cases of generic q and the isotropic (q->1) limit. Numerical solutions of the Bethe equations up to N=8 are presented. Our results are consistent with the Bethe ansatz solution being complete.
- 14 June 2014
- Published by Springer Nature
- p. 39-53
Numerical Algebraic Geometry and Differential Equations
- 14 June 2014
- Published by Springer Nature
- p. 39-53
Numerical Algebraic Geometry and Differential Equations
- 3 May 2014
- Published by Springer Nature in Rendiconti del Circolo Matematico di Palermo Series 2
- Vol. 63 (2) , 247-255
Adjunction and singular loci of hyperplane sections, II
- 3 May 2014
- Published by Springer Nature in Rendiconti del Circolo Matematico di Palermo Series 2
- Vol. 63 (2) , 247-255
Adjunction and singular loci of hyperplane sections, II