Convex Polytopes, Algebraic Geometry, and Combinatorics Laura Escobar and Kiumars Kaveh In the last several decades, convex geometry methods have The first observation is related to describing a polytope as proven very useful in algebraic geometry specifically to un- an intersection of finitely many half spaces. It appears in derstand discrete invariants of algebraic varieties. An ap- the notion of a Newton polygon introduced in Section 1 proach to study algebraic varieties is to assign to a family and is important in tropical geometry. The second observa- of varieties a corresponding family of combinatorial ob- tion is related to the construction of a polytope as a convex jects which encode geometric information about the vari- hull of finitely many points. It is central to the proof ofthe eties. Often, the combinatorial objects that arise are con- BKK theorem (see Theorem 1.2) and its generalization to vex polytopes, and convex geometry has been an essential Newton–Okounkov bodies described in Section 3. tool for this strategy. The emergence of convexity in algebraic geometry is 1. Newton Polytopes and Toric Varieties rooted in the following geometric observations. Let 풜 ⊂ The origin of appearances of convex polyhedra in algebraic ℤ푛 be a finite set, then: geometry goes back to Sir Isaac Newton and Ferdinand 1. For any vector 휉 ∈ ℝ푛, the maximum/minimum of the Minding. Newton introduced what we now know as the dot products 휉 ⋅ 푥, 푥 ∈ 풜, is attained on the boundary Newton diagram of a polynomial 푓(푥, 푡) in two variables 푥 of the convex hull of 풜. and 푡. Given an equation 푓(푥, 푡) = 0 regarded as implicitly 1 defining 푥 in terms of 푡, Newton was interested in express- 2. As 푘 → ∞ the rescaled 푘-fold sums { (푥1 + ⋯ + 푥푘) ∣ 푘 ing 푥 as an infinite series in 푡. He knew that in general 푥 ∈ 풜} converge to the convex hull of 풜. 푖 푥 = 푥(푡) may not be a power series but a series of the form ∑ 푐 푡푖/푘 for some fixed integer 푘 > 0, a series with frac- Laura Escobar is an assistant professor of mathematics at Washington Univer- 푖≥0 푖 sity in St. Louis. Her email address is [email protected]. She was supported tional exponents. In general one expects to have as many in part by NSF grant DMS-1855598. solutions as the degree of 푓 in 푥. 푎 푏 Kiumars Kaveh is an associate professor of mathematics at the University of Let 푓(푥, 푡) = ∑푎,푏 푐푎,푏푥 푡 . Consider the Newton dia- Pittsburgh. His email address is [email protected]. He was partially supported gram (or Newton polygon) of 푓(푥, 푡) to be the lower convex 2 by NSF grant DMS-1601303, a Simons Foundation Collaboration Grant for hull of {(푎, 푏) ∣ 푐푎,푏 ≠ 0} ⊂ ℝ . This is the lower part of the Mathematicians, and a Simons Fellowship. boundary of the convex hull stretching from the leftmost Communicated by Notices Associate Editor Stephan Ramon Garcia. to the rightmost point. Suppose the slopes of the (nonver- For permission to reprint this article, please contact: tical) line segments in the Newton diagram are 휇1, … , 휇푠. [email protected]. Newton showed that the fractional exponents for the first DOI: https://doi.org/10.1090/noti2137 terms in the series representations of solutions for 푥 are 1116 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 8 푠 훼푖 finite-dimensional vector space 퐿풜 = {푓(푥) = ∑푖=0 푐푖푥 ∣ ∀푖, 푐푖 ∈ ℂ} consisting of Laurent polynomials with expo- nents from 풜. The convex hull of 풜, which we denote Δ풜, is the Newton polytope of a generic element of 퐿풜. Theorem 1.2 (BKK). For a generic choice of 푓1, … , 푓푛 ∈ 퐿풜, ∗ 푛 the number of solutions 푥 in (ℂ ) of the system 푓1(푥) = ⋯ = 푓푛(푥) = 0 is the same and is equal to 푛! vol(Δ풜), where vol denotes the standard Lebesgue measure in ℝ푛. Figure 1. Example from Newton’s letter to Oldenburg dated October 24, 1676. Remark 1.3. In fact, the above form of this theorem is due to Kushnirenko. Khovanskii has found many differ- the −휇푖. This observation is a key step in proving the well- ent proofs for this theorem (worthy of Guinness Book of known Newton–Puiseux Theorem, which describes the al- Records?). Extending the BKK theorem, he also found for- gebraic closure of the field of formal power series ℂ((푡)) mulas in terms of the Newton polytope Δ풜 for genera and (see [MS15, Theorem 2.1.5]). Euler characteristics of subvarieties defined by 푓1(푥) = ⋯ = Motivated by Newton’s classic work, Vladimir Arnold 푓푘(푥) = 0, for 푘 ≤ 푛, and, as before, generic elements asked his students to work on generalizations and ana- 푓푖 ∈ 퐿풜 (see [Hov78]). logues of the notion of Newton diagram/polygon in sev- 푠 eral variables. This resulted in the modern definition of Consider the subset 푇풜 ⊂ ℂℙ defined by the Newton polytope of a polynomial. Unlike the Newton 훼0 훼푠 ∗ 푛 푇풜 = {(푥 ∶ ⋯ ∶ 푥 ) ∣ 푥 ∈ 푇 = (ℂ ) }. diagram, which is a local concept related to multiplicities of roots of a polynomial, the Newton polytope (Defini- The subset 푇풜 is isomorphic to an algebraic torus. One can 푛 tion 1.1) is a global concept which can be thought of as a show that if the differences of elements in 풜 generate ℤ , ∗ 푛 푠 refinement of the notion of degree of a polynomial. then 푇풜 is isomorphic to 푇 = (ℂ ) . Let 푋풜 ⊂ ℂℙ be the ∗ 푛 푠 Throughout we denote the multiplicative group of closure of 푇풜. Note that the torus 푇 = (ℂ ) acts on ℂℙ nonzero complex numbers by ℂ∗. The product (ℂ∗)푛 of by ∗ 훼0 훼푠 푛 copies of ℂ is an algebraic group often called an alge- 푥 ⋅ (푧0 ∶ ⋯ ∶ 푧푠) = (푥 푧0 ∶ ⋯ ∶ 푥 푧푠), 1 푛 braic torus (it contains the usual topological torus (푆 ) and the variety 푋풜 is the closure of the orbit of (1 ∶ ⋯ ∶ 1). as a maximal compact subgroup). The algebra of regu- Recall that the degree of an 푛-dimensional projective vari- ∗ 푛 lar functions on (ℂ ) is the algebra of Laurent polynomials ety 푋 ⊂ ℂℙ푠 is equal to the number of intersection points ± ± ℂ[푥1 , … , 푥푛 ]. We will use the multi-index notation, and for of 푋 with a generic plane of codimension 푛. This notion 푛 훼 훼 = (푎1, … , 푎푛) ∈ ℤ we write 푥 to denote the monomial generalizes the degree of a polynomial and provides a mea- 푎 푎 1 푛 푠 푥1 ⋯ 푥푛 . surement for how complex the embedding of 푋 in ℂℙ is. The BKK theorem can be restated as giving a formula for Definition 1.1. Let 푓(푥) = ∑ 푐 푥훼 be a Laurent polyno- 훼 훼 the degree of 푋 ⊂ ℂℙ푠. mial. The Newton polytope of 푓 is the convex hull of the 풜 finite set {훼 ∣ 푐훼 ≠ 0}. It is a convex polytope with vertices Theorem 1.4 (Alternative statement). The degree of 푋풜 ⊂ 푛 푠 in ℤ . ℂℙ is equal to 푛! vol(Δ풜). Remark 1.5. One of the simplest proofs of the BKK theo- rem, due to Khovanskii, uses the above restatement and • the classical Hilbert theorem on the relation between the • degree of a projective variety and the leading term of its Hilbert polynomial. This proof enables a generalization • • of this theorem to arbitrary systems of equations and is the basis of the theory of Newton–Okounkov bodies (Section 3). Figure 2. Newton polytope of 푓(푥, 푦) = 푥3 + 3푥푦 + 푦2 + 1. Since 푋풜 is the closure of an orbit, it is invariant under the 푇-action. It is a 푇-toric variety in the sense of the fol- One of the major discoveries of the Moscow Newton lowing definition. polyhedra school is the celebrated BKK theorem named af- ter David Bernstein (younger brother of Joseph Bernstein), Definition 1.6. An (abstract) 푇-toric variety is an irre- Askold Khovanskii, and Anatoli Kushnirenko. Fix a finite ducible variety with an algebraic 푇-action that has a finite 푛 set of lattice points 풜 = {훼0, … , 훼푠} ⊂ ℤ and consider the number of 푇-orbits. SEPTEMBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1117 푠 푛 For 푋 a 푇-toric variety there is an open dense orbit 푈0 ⊂ 휇풜 ∶ ℂℙ → ℝ is defined by 푋. After replacing 푇 with 푇/푇0, where 푇0 is the 푇-stabilizer 푠 2 of 푈0, without loss of generality we can assume that 푈0 is |푧푖| 휇풜 ∶ (푧0 ∶ ⋯ ∶ 푧푠) ↦ ∑ ( 푠 ) 훼푖 ∈ Δ풜. isomorphic to 푇 itself. 2 푖=0 ∑푗=0 |푧푗| The degree of 푋풜 is not the only geometric information we can obtain from the polytope Δ풜. In general, there is One easily verifies that 휇 is invariant under the action a beautiful correspondence between the algebro-geometry of the topological torus (푆1)푛 ⊂ 푇 = (ℂ∗)푛. Moreover, 푠 of toric varieties and the combinatorics of convex poly- 휇풜(ℂℙ ) = 휇풜(푋풜) = Δ풜. topes. The following is an example of this correspondence. Remark 1.10. The variety 푋풜 inherits a volume form/ Theorem 1.7. There is a one-to-one correspondence between measure, called the Liouville measure, from the standard 푠 the faces of the polytope Δ풜 and the torus orbits in 푋풜. Fubini–Study metric on ℂℙ . One can directly compute the pushforward of the Liouville measure on 푋풜 to Δ풜 and To make the connection between toric varieties and con- show that the pushforward measure is (a constant multi- vex polytopes more tight, one usually assumes 푋 is a nor- ple of) the Lebesgue measure on Δ풜.