What Is the Role of Algebra in Applied Mathematics? David A. Cox hen I first encountered abstract al- science, engineeering, and operations research gebra in the late 1960s, I was drawn departments. to its power and beauty. One of the I begin with examples from geometric modeling, liberating ideas of the subject is that economics, and splines to illustrate the possible Win dealing with algebraic structures, applications of algebra. I then discuss computer you do not need to worry about what the objects algebra and conclude with remarks on the role of are; rather, it is how they behave that is important. algebra in the applied mathematics curriculum. Algebra is a wonderful language for describing the behavior of mathematical objects. Geometric Modeling, Cramer’s Rule, My fascination with algebra led me to algebraic and Modules geometry, which was then among the most ab- My first example concerns an unexpected applica- stract areas of pure mathematics. At the time, I tion of Cramer’s Rule and the Hilbert-Burch Theorem would never have predicted that twenty-five years on the structure of certain free resolutions. While later I would be writing papers with computer sci- Cramer’s Rule should be familiar, free resolutions entists, where we use algebraic geometry and com- (whatever they are) may sound rather abstract. As mutative algebra to solve problems in geometric we will see, there are questions in geometric mod- modeling. The algebra that I learned as pure and eling where these topics arise naturally. abstract has come to have significant applications. My part of the story begins with the research of What do these and other applications imply Tom Sederberg and Falai Chen on parametric curves about the relation between algebra and applied in the plane. If we are given relatively prime poly- mathematics? The purpose of this essay is to ex- nomials a(t), b(t), and c(t) of degree n, then the plore some aspects of this relation in the hope of parametric equations provoking useful discussions between pure and a(t) b(t) applied mathematics. Here, “applied mathemat- (1) x = and y = ics” includes not only what students learn in math- c(t) c(t) ematics and applied mathematics departments, describe a curve in the plane. To think about this but also the mathematics learned in computer geometrically, Sederberg and Chen [23] considered lines defined by David A. Cox is the Thalheimer Professor of Mathematics at Amherst College. His email address is dac@cs. (2) A(t)x + B(t)y + C(t)=0, amherst.edu. This article is a slightly edited version of an article that where A(t),B(t), and C(t) are polynomials depend- appeared in the Gazette des Mathématiciens (April 2005, ing on the parameter t. As we vary t, the line varies no. 104, p. 13–22). It is also to appear in a future issue of also, hence the name moving line. A moving line the Gaceta de la Real Sociedad Matemática Española. follows a parametrization if for every value of the NOVEMBER 2005 NOTICES OF THE AMS 1193 parameter, the point lies on the corresponding h1p + h2q =0, where h1 and h2 are polynomi- line. In other words, (1) is a solution of (2) for all als in t. t. When we perform this substitution and clear de- • If a, b, and c have degree n in t, then p has de- nominators, we get the equation gree µ in t and q has degree n − µ, where µ is an integer satisfying 0 ≤ µ ≤ n/2 . (3) A(t)a(t)+B(t)b(t)+C(t)c(t) ≡ 0, • Up to a constant and appropriate signs, the where ≡ indicates that equality holds for all t, i.e., polynomials a(t), b(t), and c(t) are the 2 × 2 A(t)a(t)+B(t)b(t)+C(t)c(t) is the zero polynomial. minors of the matrix What algebraic structures are being used here? R A1(t) B1(t) C1(t) Since we are working over the real numbers , we . have the polynomial ring R = R[t]. Then, moving A2(t) B2(t) C2(t) lines A(t)x + B(t)y + C(t)=0 that follow (1) corre- • The equation of the curve is given by the resul- spond to elements of the set tant {(A(t),B(t),C(t)) ∈ R3 | Resultant(p, q, t) = 0. (4) A(t)a(t)+B(t)b(t)+C(t)c(t) ≡ 0}. The moving lines p and q are called a µ-basis. Thus This is an R-submodule of the free R-module R3 . the µ-basis determines both the parametrization In commutative algebra, we say that (3) is a syzygy (via the third bullet) and the equation of the curve and that (4) is a syzygy module. (via the fourth). See [7] for the details. In the mid-1990s, Sederberg asked me if I had From the point of view of commutative algebra, seen equation (3). I said, “Yes, this equation defines the first bullet says that the syzygy module is free the syzygy module of a(t),b(t), and c(t) .” As I ex- with basis (At (i),Bi(t),Ci(t)), i =1, 2 . For anyone pected, Tom asked me what “syzygy” means. I ex- who knows the theory of modules over a principal plained that in astronomy, a syzygy refers to the ideal domain, this is no surprise. But the degree re- alignment of three celestial bodies along a straight striction in the second bullet is quite different: it or nearly straight line. The term was introduced into tells us that we are really working in the homoge- mathematics by Sylvester in 1853 [27]. These days, neous situation, where we replace a(t),b(t), and c(t) a syzygy can refer to either a polynomial relation with homogeneous polynomials a(s,t),b(s,t), between invariants or a linear relation with poly- and c(s,t) of degree n. This gives the ideal nomial coefficients as in (3). = ⊂ = R But then Tom did something unexpected: he I a(s,t),b(s,t),c(s,t) S [s,t], asked me what “module” means! At first, I was shocked that he didn’t know such a basic term, but and as shown in [6] and [7], the first two bullets then I remembered that civil engineers don’t take translate into the following free resolution of the courses in abstract algebra. (Tom’s degree is in ideal I: civil engineering, though he is now in computer sci- ence.) Vectors of polynomials occur frequently in (6) 0 → S(−n + µ) ⊕ S(−2n + µ) Tom’s research, and modules over polynomial rings A1 A2 are the natural language for discussing such ob- B1 B2 jects. Yet Tom had never heard the term “module” C1 C2 until I mentioned it to him. → This made me realize that our conception of “ap- plied algebra” may need to be enlarged. I will say abc S(−n)3 → I → 0 more about this below, but first let me finish the story of moving lines. Sederberg and Chen had the (the notation S(−n) keeps track of degrees), and insight that when two moving lines follow the pa- the third bullet implies that a, b, and c are the rametrization (1), their common point of intersec- 2 × 2 minors of the 3 × 2 matrix in (6). This is a tion describes the curve. Looking deeper, they also special case of the Hilbert-Burch Theorem (see [6]).1 noticed that there are always two moving lines While this might seem sophisticated, some parts p := A1(t)x + B1(t)y + C1(t) = 0, make perfect sense. If you take the moving line (5) q := A (t)x + B (t)y + C (t) = 0 2 2 2 1This special case was discovered by Franz Meyer in 1887 that follow the parametrization and have the fol- [19], who conjectured similar results for the syzygy mod- ule of a ,...,a ∈ S. Hilbert, in his great paper of 1890 lowing special properties: 1 n [16], created the basic theory of what we now call com- • All moving lines that follow the parametrization mutative algebra. His first application was Meyer’s con- come from polynomial linear combinations jecture, which he proved via the original version of the of p and q, i.e., moving lines of the form Hilbert-Burch Theorem. 1194 NOTICES OF THE AMS VOLUME 52, NUMBER 10 equations (5) and solve for x and y via Cramer’s propensity to save). When linearized at an equi- Rule, you get librium point, the IS-LM model gives the equations −C1 B1 B1 C1 + = o + det −C B det B C sY ar I G x = 2 2 = 2 2 , A B A B − = − o det 1 1 det 1 1 mY hr Ms M , A2 B2 A2 B2 (7) − o o det A1 C1 − det A1 C1 where s, a, m, h, I ,G,Ms ,M are positive parame- A2 −C2 A2 C2 y = = . ters (for example, Ms is the money supply, and s det A1 B1 det A1 B1 A2 B2 A2 B2 is the marginal propensity to save). The goal is to see how Y and r vary when we vary the parame- Comparing this to the original parametrization (1), ters. This is where Cramer’s Rule shines: it gives it should be no surprise that a, b, and c are the formulas for Y and r that make such questions easy 2 × 2 minors of the matrix formed by the µ-basis.