Periodic Reporting for period 1 - POSSIS (Positive Solutions in the Sciences)
Berichtszeitraum: 2025-10-01 bis 2027-09-30
Polynomial equations are fundamental across various scientific disciplines, serving as powerful tools for modeling and solving real-world problems. Often, only the positive real solutions of these equations are of interest. These include applications like the Landau equations in particle physics, Nash equilibria in game theory, steady states of biochemical reaction networks, and statistical models in phylogenetics. While methods from applied algebraic geometry have already proven successful in studying complex solutions of these polynomial systems, investigating the positive solutions of these polynomials requires a paradigm shift toward real algebraic geometry.
The primary objective of the project is to develop a method for computing the positive solutions of the Landau equations. These positive solutions are crucial because they lead to singularities in the physical region of Feynman integrals, which correspond to observable phenomena in scattering experiments.
From a high-level perspective, the project aims to build bridges between real algebraic geometry and various scientific fields. While algorithms in Real Algebraic Geometry are often associated with poor worst-case complexity, many scientific problems do not fall into this category. In fact, these algorithms can often be optimized and applied efficiently by considering the specific characteristics of these problems. The goal of the project is to provide further evidence of this phenomenon and to extend the applicability of real algebraic geometry.
The primary objective of the project is to develop a method for computing the positive solutions of the Landau equations. These positive solutions are crucial because they lead to singularities in the physical region of Feynman integrals, which correspond to observable phenomena in scattering experiments.
From a high-level perspective, the project aims to build bridges between real algebraic geometry and various scientific fields. While algorithms in Real Algebraic Geometry are often associated with poor worst-case complexity, many scientific problems do not fall into this category. In fact, these algorithms can often be optimized and applied efficiently by considering the specific characteristics of these problems. The goal of the project is to provide further evidence of this phenomenon and to extend the applicability of real algebraic geometry.
The Landau equations describe the singularities of the second Symanzik polynomial, which in turn appears as the denominator polynomial of the corresponding Feynman integral. There is therefore a strong connection between the singularities of the second Symanzik polynomial and singularities of the Feynman integral itself.
Before turning to the investigation of the Landau equations, we addressed a more fundamental question: how can one describe the region in the kinematic parameter space in which a given Feynman integral converges? It was known that if the second Symanzik polynomial is strictly copositive, that is, positive on the positive real orthant, then the associated Feynman integral converges. This region of parameter space is referred to as the Euclidean region in the physics literature. In a previous paper, we provided an effective method to characterize the Euclidean region under the assumption that the kinematic parameters are sufficiently generic, thereby excluding physically relevant situations in which some particles may be massless.
During the first month of the present project, we began to study the problem of copositivity itself. Using a classical representation theorem from real algebraic geometry, combined with techniques from toric geometry, we derived a new effective method for detecting copositivity. While this approach can be applied to determine the Euclidean region even in the case of massless particles, it is more general: it applies not only to Symanzik polynomials but to arbitrary polynomials.
After that paper was uploaded to arxiv and submitted for publication, I continued to study certificates of nonnegativity and extended them using tools from toric and tropical geometry. In parallel, I implemented an initial version of a Julia package that computes the irreducible components of the Landau discriminant that intersect the positive real orthant, that is, those giving rise to positive solutions of the Landau equations. At present, neither of these projects is in a final form, both require further investigation and research.
Before turning to the investigation of the Landau equations, we addressed a more fundamental question: how can one describe the region in the kinematic parameter space in which a given Feynman integral converges? It was known that if the second Symanzik polynomial is strictly copositive, that is, positive on the positive real orthant, then the associated Feynman integral converges. This region of parameter space is referred to as the Euclidean region in the physics literature. In a previous paper, we provided an effective method to characterize the Euclidean region under the assumption that the kinematic parameters are sufficiently generic, thereby excluding physically relevant situations in which some particles may be massless.
During the first month of the present project, we began to study the problem of copositivity itself. Using a classical representation theorem from real algebraic geometry, combined with techniques from toric geometry, we derived a new effective method for detecting copositivity. While this approach can be applied to determine the Euclidean region even in the case of massless particles, it is more general: it applies not only to Symanzik polynomials but to arbitrary polynomials.
After that paper was uploaded to arxiv and submitted for publication, I continued to study certificates of nonnegativity and extended them using tools from toric and tropical geometry. In parallel, I implemented an initial version of a Julia package that computes the irreducible components of the Landau discriminant that intersect the positive real orthant, that is, those giving rise to positive solutions of the Landau equations. At present, neither of these projects is in a final form, both require further investigation and research.
The main result obtained during this short phase of the project is a criterion for copositivity that extends a classical theorem of Pólya to arbitrary sparse polynomials. This new criterion complements existing methods for certifying copositivity, such as those based on sums-of-squares certificates or nonnegative circuit polynomials.
Our Pólya-type criterion can be implemented easily by non-experts in real algebraic geometry, as it relies only on computing certain products of polynomials and checking the signs of the resulting coefficients. Due to its simplicity, this criterion has potential applications in precluding multistationarity in biochemical reaction networks and in the study of Euclidean regions of Feynman integrals in particle physics.
Our Pólya-type criterion can be implemented easily by non-experts in real algebraic geometry, as it relies only on computing certain products of polynomials and checking the signs of the resulting coefficients. Due to its simplicity, this criterion has potential applications in precluding multistationarity in biochemical reaction networks and in the study of Euclidean regions of Feynman integrals in particle physics.