Kolchin Seminar in Differential Algebra

Academic Year 2017–2018 Last updated on January 31, 2020. Other years: 2005–2006   2006–2007   2007–2008   2008–2009   2009–2010   2010–2011   2011–2012   2012–2013   2013–2014   2014–2015   2015–2016   2016–2017   2018–2019   2019–Fall

Friday, August 25, 2017, No Seminar.

Friday, September 1, 2017, 10:15–11:30 a.m. Room 5382

Gleb Pogudin, NYU Courant Institute and CUNY Graduate Center
Algorithms for Checking Global Identifiability

The following situation arises in modeling: one has a system of differential equations with parameters and wants to determine the values of these parameters measuring unknown functions (assuming that perfect noise-free measurements are possible). Usually, some of the unknown functions are impossible or very expensive to measure, so only a subset of them is available for measurement. The property of parameters to be uniquely recoverable from measuring a subset of the unknown functions is called structural global identifiability.
In the late 1980s, it was noticed that the identifiability problem can be solved using differential elimination algorithms. In this talk, I will describe recent theoretical results in this area and new efficient algorithms.
This is joint work with Hoon Hong, Alexey Ovchinnikov, and Chee Yap.

For a review of the lecture, please click talk with slides, video-1 and video-2.

You may be interested in a related talk by Nikki Meshkat (Santa Clara University), Structural Identifiability of Biological Models, on August 31 2017 at the Symbolic/Numerial Seminar: talk with slides and video.

Friday, September 8, 2017, 10:15–11:30a.m. Room 5382

Informal Session. No scheduled seminar talk.

Informal sessions at the Kolchin Seminar are open to all and attendees may bring short presentations and questions for discussion.

The Eighth International Workshop on Differential Algebra and Related Topics (DART VIII) was held at Johannes Kepler University, Linz, Austria from September 11 to 14, 2017. Click DART VIII for details.

Friday, September 15, 2017, 10:15–11:30 a.m. Room 5382

Sylvain Carpentier, Columbia University
Rational Matrix Differential Operators and Integrable Systems of PDEs

A key feature of integrability for systems of evolution PDEs

u_t = F(u),   where u = (u₁,…,u_k)
is to be part of an infinite hierarchy of commuting generalized symmetries. In all known examples, these generalized symmetries are constructed by means of Lenard-Magri sequences involving a pair of matrix differential operators (A, B). We show that in the scalar case k=1, a necessary condition for a pair of differential operators (A, B) to generate a Lenard-Magri sequence is that the ratio L := AB⁻¹ lies in a class of operators which we call integrable and which contains all ratios of compatible Poisson (or Hamiltonian) structures. We give a sufficient condition on an integrable pair of matrix differential operators (A, B) to generate an infinite Lenard-Magri sequence when the rational matrix differential operator L := AB⁻¹ is weakly non-local. If time permits, we will generalize these results to differential-difference equations.

For a review of this lecture, please click video.

Friday, September 22, 2017, No Classes and No Seminar scheduled.

Friday, September 29, 2017, No Classes and No Seminar scheduled.

Friday, October 6, 2017, 14:15–15:45 Room 5382

Eli Amzallag, City College and Graduate Center (CUNY)
Proto-Galois Groups in Small Dimensions

Let (K, δ) be a differential field and let C be the field of constants of K. Given a differential equation δ(Y) = A Y, A∈ M_n(C(t)), one might wish to compute the corresponding differential Galois group. In 2002, Hrushovski explained an algorithm for this purpose in which a group containing the Galois group is computed as a preliminary step. In a paper published in 2015 in which he analyzed Hrushovski's algorithm, Feng referred to this group as a proto-Galois group for the equation. We discuss an improvement of previous bounds in computing a proto-Galois group. We also examine the possibilities for such a group in small dimensions with a view toward establishing better bounds for applications, illustrating our approach for n = 2. This is joint work with Andrei Minchenko and Gleb Pogudin.

Friday, October 13, 2017, 10:15–11:45 a.m. Room 5382

Vahagn Aslanyan, Carnegie Mellon University
Ax-Schanuel and Strong Minimality

The Ax-Schanuel Theorem is a differential analogue of Schanuel's Conjecture proven by Ax in 1971. It establishes a transcendence result for the solutions of the exponential differential equation y' = yx' in a differential field. I will discuss similar results for other differential equations, most importantly, for the third order, non-liinear differential equation of the modular j-invariant (this Ax-Schanuel type theorem is due to J. Tsimerman and J. Pila). I will also show how that kind of results can be used to show that certain sets in differentially closed fields are strongly minimal and geometrically trivial.
A set defined by a differential equation is called strongly minimal if every Kolchin-constructible subset is either co-finite (meaning, its complement is finite) or finite. Geometric triviality of a differential equation means that if there is an algebraic relation between finitely many solutions then there must be a relation between two of them, and this relation will normally be dictated by some functional equations. Strong minimality and geometric triviality are important concepts in model theory, and our proofs are based on model theoretic methods (I will give all necessary model theoretic preliminaries during the talk).
As an application I will give a new proof for a theorem of J. Freitag and T. Scanlon establishing strong minimality and geometric triviality of the differential equation of the j-function. If time permits, I will discuss Ax-Schanuel type conjectures for the Painlevé equations based on the results of J. Nagloo and A. Pillay on strong minimality and geometric triviality of those equations.

For a copy of the paper, please click paper.

Friday, October 13, 2017, 2:15–3:45 p.m. Room 5382

Yuri Berest, Cornell University
Differential Isomorphism and Equivalence of Algebraic Varieties

It is a basic fact of commutative algebra that an affine algebraic variety is uniquely determined by its algebra of regular functions. Following Grothendieck, one can also associate to any such variety X a canonical noncommutative algebra D(X), which is the ring of (global regular) differential operators on X. A natural question then is: To what extent does the algebra D(X) determine X ? Although not much is known about this question in higher dimensions, a fairly complete answer with many deep and surprising connections is available in the case of curves. In this talk, after reviewing some history and motivation, I will discuss recent work showing how the above question leads to interesting generalizations of some classical results in the theory of finite-dimensional linear algebraic groups to the infinite-dimensional case.

Friday, October 20, 2017, 10:15–11:30 a.m. Room 5382

Jason Bell, University of Waterloo
New Methods in Hypertranscendence

Let F(x) be a complex power series and let φ be either an endomorphism or a derivation of the ring of complex power series. We say that F(x) is hypertranscendental with respect to φ if the family {φⁿ(F(x))}_{n ≥ 0} is algebraically independent over the field of rational functions. We discuss some recent work on this problem when φ is a Mahler operator x↦ x^k.

Friday, October 27, 2017, 10:15–11:30 a.m. Room 5382

Thomas Scanlon, University of California at Berkeley
Applications of Characterizations of Skew-Invariant Varieties

In work with Medvedev, I classified the skew-invariant subvarieties of so-called split polynomial dynamical systems. Here, a split polynomial dynamical system is one of the form F : 𝔸ⁿ → 𝔸ⁿ given in coordinates as (x₁, … , x_n) ↦ (f₁(x₁), f₂(x₂), …, f_n(x_n)), where each f_i is a polynomial in one variable. The "skew" in "skew-invariant" means that we work over a field K equipped with an endomorphism σ : K → K. A subvariety V of 𝔸ⁿ is skew-invariant if F maps V to V ^σ, the transform of V under σ.
    In most applications of our theorem to date, only the case that σ is the identity is used and the resulting classification of the invariant varieties may be obtained from methods of complex dynamics, as shown by Pakovich.
    In this lecture, I will speak about two applications which make essential use of the generalization to skew-invariance: a theorem proven jointly with Medvedev and Nguyen that Mahler functions of polynomial type with respect to multiplicatively independent exponents are algebraically independent, and a project with Medvedev to extend our classification of (skew-)invariant varieties to what we call triangular dynamical systems (though what have been called skew-products in the literature): algebraic dynamical systems of the form F : 𝔸ⁿ → 𝔸ⁿ given in coordinates as (x₁, …, x_n) ↦ (f₁(x₁), f₂(x₁, x₂), …, f_n(x₁, …, x_n)), where each f_i is a polynomial in the variables x₁, … x_i.

Friday, November 3, 2017, 10:15–11:30 a.m. Room 5382

Informal Session

Friday, November 10, 2017, 10:15–11:30 a.m. Room 5382

Mike Mikalajunas, Futurion Associates
Towards a More Unified Approach to Exact Integration of Differential Equations

In this talk, we will present a specific class of multivariate functions that can be recursively defined in terms of multivariate polynomials and the differential of multivariate polynomials. A recursive differential expansion, referred to as a Multivariate Polynomial Transform, and its inverse can be applied to algebraic and differential equations. The mathematical properties of this transform could be exploited much further for establishing the basic fundamental building blocks of the "theory of everything". Under this unified approach to exact integration, common PDEs from physics and engineering can now be solved directly from their original forms without resorting to ad hoc transformation processes normally needed to reduce the PDEs to more integrable forms. The process will be illustrated with examples.
For a copy of the revised slides, please click slides.

Friday, November 17, 2017, 10:15–11:30 a.m. Room 5382

Joel Nagloo, Bronx Community College (CUNY)
Towards Strong Minimality and the Fuchsian Triangle Group

From the work of Freitag and Scanlon, we know that the ODEs satisfied by the Hauptmoduls of arithmetic subgroups of SL₂(ℤ) are strongly minimal and geometrically trivial. A challenge is to now show that the same is true of ODEs satisfied by the Hauptmoduls of all (remaining) Fuchsian triangle groups. The aim of this talk is to both explain why this an interesting/important problem and also to discuss some of the progress made so far.

Tuesday (Friday schedule), November 21, 2017, 10:15–11:30 a.m. Room 5382

Beatriz Pascual Escudero, Universidad Autónoma de Madrid
Invariants of Singularities via the Nash Multiplicity Sequence and Their Connection to Hironaka's Order Function

When we study algorithmic or constructive Resolution of Singularities, we make use of invariants that allow us to distinguish among different singular points of an algebraic variety. Attending to them, we choose the centers of a sequence of blow ups that will eventually lead to a resolution of the singularities of the initial variety.

On the other hand, arc spaces are useful in the study of singularities, since they detect properties of algebraic varieties, including smoothness. They also let us define numerous invariants. In particular, the Nash multiplicity sequence is a non-increasing sequence of positive integers attached to an arc in the variety which stratifies the arc space. As we will see, this sequence gives rise to a series of invariants of singularities which turn out to be strongly related to those that we use for constructive resolution of singularities for varieties defined over fields of characteristic zero. Moreover, these invariants defined by means of arc spaces do not rely on the peculiarities of the characteristic zero case, so they pose interesting questions for the case of varieties defined over perfect fields, regardless of their characteristic.
For a review of the talk, please click video.

Friday, November 24, 2017, Thanksgiving Weekend (No Seminar).

Friday, December 1, 2017, 10:15–11:30 a.m. Room 5382

Informal Session

Friday, December 8, 2017, 10:15–11:30 a.m. Room 5382

Joel Nagloo, Bronx Community College (CUNY)
Towards Strong Minimality and the Fuchsian Triangle Group, Part II

From the work of Freitag and Scanlon, we know that the ODEs satisfied by the Hauptmoduls of arithmetic subgroups of SL₂(ℤ) are strongly minimal and geometrically trivial. A challenge is to now show that the same is true of ODEs satisfied by the Hauptmoduls of all (remaining) Fuchsian triangle groups. The aim of this talk is to both explain why this an interesting/important problem and also to discuss some of the progress made so far.

Friday, February 2, 2018, 10:15–11:30 a.m. Room 5382

No seminar (Mathfest Day)

Friday, February 9, 2018, 14:45–15:45, Room 5382

Erdal Imamoglu, North Carolina State University
Algorithms for Solving Linear Differential Equations with Rational Function Coefficients

We present two algorithms for computing hypergeometric solutions of a second order linear differential equation with rational function coefficients. Our first algorithm uses quotients of formal solutions, modular reduction, Hensel lifting, and rational reconstruction. Our second algorithm first tries to simplify the input differential equation using integral bases for differential operators and then uses quotients of formal solutions.
For a review of the talk, please click video.

Friday, February 16, 2017, 10:30–11:30 a.m. Room 5382

Michael Wibmer, University of Pennsylvania
Algebraic Theory of Difference Equations

Difference equations are a discrete analog of differential equations. This talk will explain how the algebraic theory of difference equations provides a way to tackle problems in number theory, discrete dynamical systems and the theory of differential equations. For example, I have used the Galois theory of difference equations to prove a special case of the dynamical Mordell-Lang conjecture. Finiteness results will be a guiding theme for this talk. In particular, we will show that the ideal of all difference algebraic relations among the solutions of a linear differential equation is finitely generated.

Friday, February 23, 2018, 14:30–15:30, Room 5382

Peter Olver, University of Minnesota
Algebras of Differential Invariants

A classical theorem of Lie and Tresse states that the algebra of differential invariants of a Lie group or (suitable) Lie pseudo-group action is finitely generated. I will present a fully constructive algorithm, based on the equivariant method of moving frames, that reveals the full structure of such non-commutative differential algebras, and, in particular, pinpoints generating sets of differential invariants as well as their differential syzygies. A variety of applications and outstanding issues will be discussed, including recent applications to classical invariant theory, equivalence and symmetry detection in image processing, and some surprising results in classical surface geometries.
For a copy of the slides, please click slides.
For a review of the lecture, please click video.

Friday, March 2, 2018, 14:15–15:30, Room 5382

Anne Shiu, Texas A & M University
Identifiability of Linear Compartment Models: The Singular Locus

This talk addresses the problem of parameter identifiability—that is, the question of whether parameters can be recovered from data—or linear compartment models. Using standard differential algebraic techniques, the question of whether such a model is (generically, locally) identifiable is equivalent to asking whether the Jacobian matrix of a certain coefficient map, arising from input-output equations, is generically full rank. A formula for these input-output equations was given recently by Meshkat, Sullivant, and Eisenberg. Here we build on their results by giving a formula for the resulting coefficient maps. This formula is in terms of acyclic subgraphs of the directed graph underlying the linear compartment model. As an application, we prove that two families of linear compartment models—cycle and mammillary (star) models—are identifiable. We accomplish this by determining the defining equation for the singular locus of non-identifiable parameters. Our work helps to shed light on the open question of which linear compartment models are identifiable. This is joint work with Elizabeth Gross and Nicolette Meshkat.
For a copy of slides, please click slides.
For a review of the talk, please click video.

Friday, March 9, 2018, 10:15–11:30 a.m. Room 5382

Gleb Pogudin, New York University
Analyticity of Power Series Solutions of Algebraic ODEs

One natural approach to solving a system of algebraic differential equations is to find a formal power series satisfying the system. It has been known for years that such a formal power series might diverge in any neightborhood, so might not result in an analytic solution. Thus, the question is how to check whether all formal the power series solutions of a given system do converge. Surprisingly, there exists a purely algebraic sufficient criteria for that, which was obtained recently by O. Gerasimova and Yu.P. Razmyslov. It is based on the transcendence degree of the corresponding differential algebra and its homomorphic images. In the talk I will describe these results and explain how these results explain the fact that many power series solutions of equations arising in practice actually define analytic functions.
For a review of the talk, please click video.

Friday, March 16, 2018, 10:15–11:30 a.m. Room 5382

William Sit, The City College of New York
What Initial Values Guarantee Existence and Uniqueness in Algebraic Differential Systems?

This talk is complementary to the talk given last week by Gleb Pogudin on the convergence of power series solutions to systems of ODEs. The main result reported by Pogudin is that under certain conditions, for any prescribed initial values, the formal power series solution, if it exists, will be convergent. Pritchard and Sit studied initial value problems for differential algebraic ODES in relation to the question of existence and uniqueness of solutions. In this talk, we describe a theoretical as well as computational approach to obtain algebraic constraints on initial values that would guarantee existence and uniqueness of solutions. Our method works directly for a wide class of linear or non-linear systems of first order ordinary differential equations and in addition to obtaining the constraints, the algorithm (implemented in Axiom) will also provide equivalent systems where the first derivatives of the dependent variables are explicitly given in terms of rational functions. These rational vector fields can be integrated in a numerically stable way. Higher order systems can be dealt with by the usual reduction to first order systems. Singularities are exposed by the algorithm in some simple examples.
For a revised and shortened version of the slides, please click slides.
Reference: F. L. Pritchard and W. Sit, "On Initial Value Problems for Ordinary Differential-Algebraic Equations," in Grobner Bases in Symbolic Analysis, M. Rosenkranz and D. Wang eds., Radon Series Comp. Appl. Math 2, 283-340, Walter de Gruyter, 2007.

Friday, March 23, 2018, 10:15–11:30 a.m. Room 5382

Francis Valiquette, SUNY at New Paltz
Group Foliation of Finite Difference Equations Using Equivariant Moving Frames

Given a finite difference equation with continuous symmetry group G, I will explain how to solve such an equation using the method of equivariant moving frames.
For a copy of the slides, please click slides.
For a review of the talk, please click video.

Friday, March 23, 2018, 14:00–15:30, Room 5382

Eli Amzallag, City University of New York
Galois Groups of Differential Equations and Representing Algebraic Sets

The algebraic framework for capturing properties of solution sets of differential equations was formally introduced by Ritt and Kolchin. As a parallel to the classical Galois groups of polynomial equations, they devised the notion of a differential Galois group for a linear differential equation. Just as solvability of a polynomial equation by radicals is linked to the equation’s Galois group, so too is the ability to express the solution to a linear differential equation in closed form linked to the equation’s differential Galois group. It is thus useful even outside of mathematics to be able to compute and represent these differential Galois groups, which can be realized as linear algebraic groups; indeed, many algorithms have been written for this purpose. The most general of these is Hrushovski’s algorithm and so its complexity is of great interest. A key step of the algorithm is the computation of a group called a proto-Galois group, which contains the differential Galois group. As a proto-Galois group is an algebraic set and there are various ways to represent an algebraic set, a natural matter to investigate in this regard is which representation(s) are expected to be the "smallest".
Some typical representations of algebraic sets are equations (that have the given algebraic set as their common solutions) and, for the corresponding ideal, Gröbner bases or triangular sets. In computing any of these representations, it can be helpful to have a degree bound on the polynomials they will feature based on the given differential equation. Feng gave such a bound for a Gröbner basis for a proto-Galois group’s ideal in terms of the size of the coefficient matrix. I will discuss how my joint work with Andrei Minchenko and Gleb Pogudin improved this bound by focusing on equations that define such a group instead of its corresponding ideal. This bound also produces a smaller degree bound for Gröbner bases than the one Feng obtained. Recent work by M. Sun shows that Feng’s bound can also be improved by replacing Feng's uses of Gröbner bases by triangular sets. Sun’s bound relies on my joint work with Pogudin, Sun, and Ngoc Thieu Vo on the complexity of triangular representations of algebraic sets, work that more generally suggests using triangular sets in place of Gröbner bases to potentially reduce complexity. I will explain why this is the case.

Friday, March 30 and April 6, 2018,

Spring Recess, No Seminar

Wednesday (Friday schedule), April 11, 2018, 10:15–11:30 a.m. Room 5382

No Seminar.

Friday, April 13, 2018, 10:15–11:30 a.m. Room 5382

No seminar.

Friday, April 20, 2018, 10:00–11:15 a.m. Room 5382

André Platzer, Carnegie Mellon University
Differential Equation Axiomatization

We prove the completeness of an axiomatization for differential equation invariants, so all true invariants are provable. First, we show that the differential equation axioms in differential dynamic logic are complete for all algebraic invariants. Our proof exploits differential ghosts, which introduce additional variables that can be chosen to evolve freely along new differential equations. Cleverly chosen differential ghosts are the proof-theoretical counterpart of dark matter. They create new hypothetical states, whose relationship to the original state variables satisfies invariants that did not exist before. The reflection of these new invariants in the original system enables its analysis.
We then show that extending the axiomatization with existence and uniqueness axioms makes it complete for all local progress properties, and further extension with a real induction axiom makes it complete for all real arithmetic invariants. This yields a parsimonious axiomatization, which serves as the logical foundation for reasoning about invariants of differential equations. Moreover, our approach is purely axiomatic, and so the axiomatization is suitable for sound implementation in foundational theorem provers.
This talk is based on joint work with Yong Kiam Tan at LICS, 2018.
For a review of this talk please click video2. For a review of the talk on Thursday, April 19, 2018, at The Joint CUNY Graduate Center-Courant Seminar in Symbolic-Numeric Computing, please try (may not work) video1.

Friday, April 27, 2018, 10:15–11:30 a.m. Room 5382

Boris Adamczewski, Université Lyon I
Mahler's Method: Old and New

Any algebraic (resp. linear) relation over the field of rational functions with algebraic coefficients between given analytic functions leads by specialization to an algebraic (resp. linear) relation over the field of algebraic numbers between the values of these functions at any algebraic point (where of course the functions are well-defined). Number theorists have long been interested in proving results going in the other direction. Though the converse result is known to be false in general, there are few known instances where it essentially holds true. In each case, an additional structure is required: the analytic functions under consideration do satisfy some kind of differential/difference equation. Mahler's method, which was initiated by Mahler at the end of 1920s, provides an instance of such a situation. In this talk, I will discuss recent results by Philippon on the one hand, and by Faverjon and the speaker on the other hand which solve, partly or completely, some old problems of this theory. The discussion will include the case of Mahler's method in several variables, as well as applications related to automata theory.
For a review of the talk, please click video.

Friday, May 4, 2018, 10:15–11:30 a.m. Room 5382

Dmitry A. Lyakhov, KAUST
Algorithmic Lie Symmetry Analysis and Group Classification for Ordinary Differential Equations

Sophus Lie proposed a systematic method for solving nonlinear differential equations. Typically it leads to a large system of overdetermined partial differential equations and relies on tools of differential algebra. We will show some recent advances in this field and raise open questions on algorithmic group classification.

For a review of the lecture, please click video.

Friday, May 11, 2018, 10:15–11:30 a.m. Room 5382

Amaury Pouly, Max Planck Institute, Saarbrücken
A Truly Universal Polynomial Differential Equation

Lee A. Rubel proved in 1981 that there exists a universal fourth-order algebraic differential equation

P(y', y'', y''', y'''') = 0    (1)
and provided an explicit example. It is universal in the sense that for any continuous function f : ℝ → ℝ and any continuous positive function ε : ℝ → (0, +∞), there exists an infinitely smooth solution y to (1) such that ∣ f(t)-y(t) ∣ ≤  ε(t) for all real t. In other words, there is always a solution of (1) that is ε-close to f everywhere. However this result suffers from a big shortcoming, identified as an open question by Rubel in his paper: "It is open whether we can require that the solution that approximates f be the unique solution for its initial data." Indeed, the solution to (1) is not unique when one adds the condition that y(0)=y₀ for example. In fact, one can add a finite number of such conditions and still not get uniqueness of the solution. This is not surprising because the construction of Rubel fundamentally relies on the non-uniqueness of the solution to work.
Our main result is to answer Rubel's question positively. More precisely, we show that there is a polynomial P of order k such that for any continuous function f : ℝ → ℝ and any continuous positive function ε : ℝ → (0, +∞), there exist real y ₀, y'₀, …, y₀^(k-1)  such that the unique analytic solution y to
P(y, y', …, y^(k))=0 and y(0)=y ₀, y'(0)=y'₀, … , y^(k-1)(0)=y₀^(k-1)  is such that ∣ f(t)-y(t) ∣ ≤  ε(t) for all real t. The major difference with Rubel's result is that we get unicity by requiring that the solution be analytic. This requires a completely different and more involved construction. A by-product of the proof is that y₀ is constructive and in fact computable from f and ε.
For a copy of the slides, please click slides. For a review of the lecture, please click video.

Friday, May 11, 2018, 14:00–15:00 Room 5382

Julio R. Banga, Marine Research Institute, Vigo
Optimality Principles and Identification of Dynamic Models of Biosystems

Deterministic nonlinear ODEs are widely used in computational systems biology. The inverse problem of parameter estimation can be very challenging and has received great attention. Here, we will discuss a generalization of this problem which addresses the following question: given a dynamic model and observed time-series data, estimate the time-invariant model parameters, the unmeasured time-dependent inputs and the underlying optimality principle that is consistent with the measurements.
We will justify the use of optimality principles in this context, and we will present an inverse optimal control formulation that can be used to solve the above problem under some assumptions. We will also discuss associated identifiability issues, and how to surmount them in special cases. We will illustrate these ideas with examples of increasing complexity involving metabolic and signaling pathways.
For a copy of the slides, please click slides. For a review of the lecture, please click video.

Friday, May 11, 2018, 15:00–16:00 Room 5382

Hoon Hong, North Carolina State University
Checking Consistency of Algebraic Inequalities by Prolongation and Linear Relaxation

The combination of prolongation and relaxation is a fundamental technique for reducing a difficult problem to a larger but easier problem. It has been successfully applied to checking the consistency of polynomial equations over complex numbers by Sylvester and Macaulay, reducing it to that of linear equations. It has been also successfully applied to differential equations by Kolchin, reducing it to that of polynomial equations. In this talk, we share our on-going effort on applying the technique to polynomial equations/inequalities over real numbers, reducing it to that of linear equations/inequalities. This is a joint work with Brittany Riggs.