ADJUNCTION INEQUALITY FOR REAL ALGEBRAIC CURVES

G.MIKHALKIN

Abstract. The zero set of a real polynomial in two variable is a curve in ℝ^{2}. For a generic choice of its coefficients this is a non-singular curve, a collection of circles and lines properly embedded in ℝ^{2}. What topological arrangements of these circles and lines appear for the polynomials of a given degree? This question arised in the 19th century in the works of Harnack and Hilbert and was included by Hilbert into his 16th problem. Several partial results were obtained since then (see [5], [1], [7], [8]). However the complete answer is known only for polynomials of degree 5 (see [6]) or less. The paper presents a new partial result toward the solution of the 16th Hilbert problem.

A consequence of the main theorem of the paper implies that the zero set of a polynomial of degree 7 can not be isotopic to a curve pictured on Fig.1 (contrary to the curve pictured on Fig.2). The realizibility of Fig.1 does not contradict to the previously known restrictions. The proof of the main theorem makes use of the proof by Kronheimer and Mrowka [3] of the Thom conjecture in ℂP^{2}.

Figure 1. This picture is not isotopic to a curve of degree 7.

Figure 2. This picture is isotopic to a curve of degree 7.

1. Statement of the theorem

Let f and g be two real homogeneous polynomials in two variables of degrees m and n, m ≥ n, m ≡ n (mod 2). Let

ℝA = {(x_{0} : x_{1} : x_{2}) ∈ ℝP^{2} | f(x_{0}, x_{1}, x_{2}) = 0}
ℝB = {(x_{0} : x_{1} : x_{2}) ∈ ℝP^{2} | g(x_{0}, x_{1}, x_{2}) = 0}

Research at MSRI supported in part by NSF grant #DMS 9022140.

be the corresponding curves in ℝP^{2}. Let

ℂA = {(x_{0} : x_{1} : x_{2}) ∈ ℂP^{2} | f(x_{0}, x_{1}, x_{2}) = 0}
ℂB = {(x_{0} : x_{1} : x_{2}) ∈ ℂP^{2} | g(x_{0}, x_{1}, x_{2}) = 0}

be their complexifications in ℂP^{2}. If the coefficients of f and g are generic then ℂA and ℂB are transverse smooth submanifolds of ℂP^{2}. We call such curves nonsingular. The real parts ℝA and ℝB of non-singular curves are transverse smooth submanifolds of ℝP^{2}.

The complex conjugation involution conj : (x_{0} : x_{1} : x_{2}) → (x̄_{0} : x̄_{1} : x̄_{2}) acts on ℂP^{2}, ℂA and ℂB fixing ℝP^{2}, ℝA and ℝB. Note that ℂA − ℝA is either conneceted (then ℝA is called of type II) or consists of two components interchanged by conj (then ℝA is called of type I) since ℂA/ conj is connected and ∂ℂA/ conj = ℝA. If ℝA is of type I then following [7] we equip ℝA with complex orientation, i.e. with the boundary orientation induced from one of the components of ℂA − ℝA. Because of ambiguity in the choice of the component of ℂA − ℝA this orientation is defined up to simultaneous reversion of the orientations of all the components of ℝA.

Let C ⊂ ℝP^{2} be an oriented null-homologous topologically embedded (multicomponent) curve. Following [8] we define a locally constant function called the index

| ind | : ℝP^{2} − C → ℤ

in the following way. The index of the non-orientable component of ℝP^{2} − C is zero. The complement of the non-orientable component is a disjoint union of disks. A regular neighborhood N of each of these disks is ℝ^{2} and in N there is a well-defined index of a point with respect to C ∩ N once we pick an orientation of N. We define ind |_{N−C} as the absolute value of the index in N.

Figure 3. Function | ind |.

For each locally constant function F : ℝP^{2} − C → ℝ we define

∫_{ℝP^{2}} F dχ = ∑_{G} F (G)χ(G)

over all components G of ℝP^{2} − C (χ denotes the Euler characteristic).

Let ℝA and ℝB be the curves of type I equipped with the complex orientations. We associate the following curve C to ℝA and ℝB. Take ℝA ∪ ℝB and smooth all the points of intersection according to their orientations choosing any pattern of Fig.4 (cf. smoothing in [4]). (Recall that ℝA and ℝB are transverse since we assume that coefficients of f and g are generic and C is homologous to zero since m ≡ n (mod 2)).

Figure 4. Smoothings of a double point.

Let G be a domain in ℝP^{2} such that ∂G is a component of C. Let c_{out} (c_{in}) be the number of the outward (inward) with respect to G cusps on the boundary of G. Let χ′(G) = χ(G) − \frac{1}{2}(c_{out} − c_{in}) (cf. Remark 1.3 of [4]). Define ”the negative variation” of χ′ by the following rules. For a connected G let χ′_{−}(G) = −χ′(G) if χ′(G) < 0 and χ′_{−}(G) = 0 if χ′(G) ≥ 0. For a disconected G define χ′_{−}(G) as the sum of χ′ over all components of G. Define

Ξ = ∑_{s=1}^{∞} sχ′_{−}{| ind(x)| > s}.

Note that Ξ = 0 if | ind | < 2 on all components of ℝP^{2} − C of negative Euler characteristic. This will be the case for our applications. Let d be the number of the intersection points of ℝA and ℝB.

Theorem 1.

∫_{ℝP^{2}} | ind |dχ ≤ \frac{(m + n)^{2}}{4} − 3n + d + Ξ.

Note that there is ambiguity in choosing the complex orientations of ℝA and ℝB and in smoothing of double points of ℝA ∪ ℝB. The inequality of the theorem holds for any choice of the complex orientation and smoothings.

Remark . If ℝA is a non-singular curve of type I of degree m then the curve obtained by taking s parallel copies of each component of ℝA is isotopic to a non-singular curve of type I of degree sm. Indeed, if f defines ℝA then

f(f − ϵh) … (f − (s − 1)ϵh) = δp

defines the requred curve for a choice of real numbers ϵ, δ and polynomials h and p of degree m and sm.

This observation combined with the theorem yields additional restrictions on mutual position of two curves of type I. In particular, it allows to use the theorem in the case when the degrees of two curves are of opposite parity.

2. Proof of Theorem 1

Consider first the special case when ℝA ∩ ℝB = ∅. Let A_{+} be the component of ℂA − ℝA which determines the chosen complex orientation of ℝA and let B_{−} be the component of ℂB − ℝB which determines the orientation opposite to the chosen orientation of ℝB. Let

F′ = A_{+} ∪ B_{−} ∪ L,

where L = ⋃_{s=1}^{∞}{| ind | ≥ s}. Note that F′ makes an integer cycle such that the orientation induced from F′ on A_{+} is holomorphic and the orientation induced from F′ on B_{−} is antiholomorphic.

Let F be the surface smoothly generically immersed into ℂP^{2} obtained by smoothing the corners and, then, perturbation of F′. The double points of F come from two sources, from A_{+} ∩ B_{−} and from perturbation of ⋃_{s=1}^{∞}{| ind | ≥ s}. Note that the double points coming from A_{+} are all negative since all points of intersection of ℂA and ℂB equipped with the holomorphic orientation are positive.

To obtain F out of F′ we perturb every component G′ ⊂ ℝP^{2} of {| ind | ≥ s}, s > 1, to G along a generic vector field v normal to G′ but tangent to ℂA∪ℂB along ∂G′ ⊂ ℝA ∪ ℝB (to keep F a cycle). The field iv is tangent to G′ and to ∂G′. The zeroes of v and iv are of opposite index. Therefore (inductively by s) we can choose the perturbation field v so that G (equipped with the orientation induced from F) intersects the result of perturbation of {| ind | ≥ t}, t < s, in |χ(G′)| points, which are negative if χ(G′) > 0 and positive if χ(G′) < 0. Therefore, the total number of positive double points of F is Ξ.

Note that

[F] = \frac{m − n}{2}[ℂP^{1}] ∈ H_{2}(ℂP^{2}).

since F ∪ conj(F) is homologous to [ℂA] − [ℂB] = (m − n)[ℂP^{1}] and conj acts as (−1) on H_{2}(ℂP^{2}). We need the following strengthening of the Thom conjecture [3]

−χ(F) + 2Ξ ≥ \frac{(m − n)^{2}}{4} − 3\frac{m − n}{2}. (1)

This strengthening is implicitly contained in [3]. Indeed, it is proven there that if H is a surface smoothly embedded to ℂP^{2}#ℂ̄P^{2}# … #ℂ̄P^{2} and H.H ≥ 0 then −χ(H) ≥ H.H + K.H, where K = (−3, 1, …, 1) ∈ H_{2}(ℂP^{2}#ℂ̄P^{2}# … #ℂ̄P^{2}) is the canonical class. We can get H out of F by blowing up all the double points of F. The intersection of F with a small sphere centered at a double point of F is the Hopf link (the difference between the positive and negative Hopf links disappears if we forget the orientation of the links). The resulting H intersects each exceptional divisor at two points, which are of the same sign (which is positive) for positive double points and of the opposite signs for negative double points. Therefore, H.H = F.F −4Ξ and K.H = −3\frac{m−n}{2} +2Ξ while χ(H) = χ(F). This implies (1). Note that the assumption m ≥ n insures H.[ω] > 0 and thus enables us to apply [3].

Figure 5. A double point in the real plane and its link in the complexification.

By additivity of χ

(2) χ(F) = χ(A_{+}) + χ(B_{−}) + χ(L) =

χ(A_{+}) + χ(B_{−}) + ∑_{s=1}^{∞} χ(| ind | ≥ s) = χ(A_{+}) + χ(B_{−}) + ∫_{ℝP^{2}} | ind |dχ.

But χ(A_{+}) = \frac{1}{2}χ(ℂA) = −\frac{m^{2}−3m}{2} and χ(B_{−}) = \frac{1}{2}χ(ℂB) = −\frac{n^{2}−3n}{2} by the adjunction formula for the holomorphic curves ℂA and ℂB. Combining this with (1) and (2) we get

∫_{ℝP^{2}} | ind |dχ ≤ \frac{m^{2} − 3m}{2} + \frac{n^{2} − 3n}{2} − \frac{(m − n)^{2}}{4} +3\frac{m − n}{2} +2Ξ = \frac{(m + n)^{2}}{4} −3n+2Ξ.

We return to the general case when ℝA and ℝB intersect in d points. We do the same thing still starting from F′ = A_{+} ∪ B_{−} ∪ ⋃_{s=1}^{∞}{| ind | ≥ s} but we have to pay special attention to smoothing of F′ near ℝA ∩ ℝB. The intersection of A_{+}, B_{−} and ℝP^{2} with a small sphere centered in a point of ℝA ∩ ℝB is pictured on Fig.5. It consists of 6 arcs, 2 of them, a and b come from the intersection with A_{+} and B_{−} and 4 of them, c, d, e, f, come from the intersection with ℝP^{2}. For perturbation of F′ we need to choose such a framing on the circle a∪d∪b∪f that the result of perturbation of a ∪ d ∪ b ∪ f along this framing is not linked with c ∪ d ∪ e ∪ f. The number of (full right) twists j of the framing along f has to be equal to the number of twists along d in order for these circles to be homologically unlinked. These circles are not linked if (and only if) j = 0 or 1 (see Fig.6). This corresponds to the framing obtained by multiplication by i of the vector field tangent to the result of smoothing with two inwards or outwards cusps (cf. Fig.4). Gluing the disk bounded by a ∪ d ∪ b ∪ f along

Figure 6. Choosing the framing along d and f. the arcs f and d decreases the Euler characteristic of F by 1, so the number of the points of ℝA ∩ ℝB appears in the final formula.

3. Applications

By the Harnack inequality [2] an affine curve of degree 2k + 1 has no more then k(2k − 1) ovals.

Proposition 1. If an affine curve of degree 2k + 1 consists of an arc and k(2k − 1) ovals with disjoint interiors then each component of the complement of the arc in the affine plane contains at least \frac{(k−1)(k−2)}{2} ovals.

Remark . This proposition is sharp in the following sense. For any k > 0 there exist affine curves of degree (2k + 1) which consist of k(2k − 1) ovals and an arc where the arc separates \frac{(k−1)(k−2)}{2} ovals from \frac{(3k−2)(k+1)}{2} ovals. Construction of such curves can be easily extracted from Harnack’s construction of curves with (k(2k − 1) ovals [2].

Proof. We apply Theorem 1 to the union of the affine curve and the infinite line in ℝP^{2}. The projectivization of the affine curve is of type I because the curve is extremal for the Harnack inequality (see [7]). Suppose the arc separates A ovals from k(2k − 1) − A ovals. Then for a proper choice of the orientation of the infinite line and the affine curve

∫_{ℝP^{2}} | ind |dχ ≥ k(2k − 1) − A + 1 − A

(the right hand side is the Euler characteristic of the region with | ind | = 1, the only possible regions with | ind | > 1 are the interiors of some of the A ovals (this depends on their orientation), d = 1 and Ξ = 0. Theorem 1 implies now that 1 + k(2k − 1) − 2A ≤ (k + 1)^{2} − 3 + 1 and, therefore A ≥ \frac{(k−1)(k−2)}{2}.

Corollary 1. The curve pictured on Fig.1 is not isotopic to an affine algebraic curve of degree 7.

Remark . Degree 7 is the smallest degree when such a restriction appears. For an affine curve of degree 5 which consists of an arc and some ovals there are no constraints on distribution of the ovals between the half-planes (as long as their total number is no more than 6).

For an even degree 2k Harnack’s construction (for the maximal number of ovals) produces a curve ℝP^{2} which consists of a non-empty oval, \frac{k^{2}−3k+2}{2} empty ovals inside the non-empty ovals and \frac{3k^{2}−3k}{2} empty ovals outside of the non-empty oval (see Fig.8 for k = 4). Let ℝA ⊂ ℝP^{2} be a curve of degree 2k with such an arrangement of the ovals.

Proposition 2. Not more than (k−2) out of the \frac{k^{2}−3k+2}{2} interior ovals of ℝA may be contained inside an ellipse (a non-singular curve of degree 2) in ℝP^{2} not intersecting ℝA.

Remark . This proposition is also sharp. For the Harnack curves of degree 2k we may find ellipses enclosing (k − 2) of the interior ovals and not intersecting the curves.

Figure 7. Orientations of the ellipse and the non-empty oval of ℝA.

Proof. Let L > 0 be the number of the interior ovals inside the ellipse. Apply Theorem 1 to ℝA and the ellipse oriented so that the non-empty oval of ℝA and the ellipse are oriented in the opposite way (see Fig.7). We get

∫_{ℝP^{2}} | ind |dχ ≥ \frac{3k^{2} − 3k}{2} − \frac{k^{2} − 3k + 2}{2} + 2L = k^{2} − 1 + 2L,

d = 0 and Ξ = 0. Therefore, k^{2} − 1 + 2L ≤ (k + 1)^{2} − 6 and L ≤ k − 2.

Corollary 2. It is not possible to separate the 3 interior ovals of a curve of degree 8 with topological arrangement < 18 ⊔ 1<3>> from the rest of the ovals by a conic.

Figure 8. The Harnack curve of degree 8 and a conic.

Remark . Theorem 1 generalizes for curves on algebraic surfaces other than ℝP^{2}. Instead of m ≡ n (mod 2) we require that the union of the two curves is nullhomologous in the surface (so that | ind | is well-defined). The formula then turns to

∫_{ℝP^{2}} | ind |dχ ≤ \frac{([ℂA] + [ℂB])^{2}}{4} + K.[ℂB] + d + Ξ,

where K is the canonical class of the complexification of the surface.

If the geometric genus of the surface is positive then we do not need to require that the degree of ℂA is greater than or equal to the degree of ℂB.

References

[1] Arnold, V.I.: On the distribution of the ovals of real plane curves, involutions of 4-dimensional smooth manifolds and the arithmetics of integer-valued quadratic forms. Functsional’ni Analiz i ego Prilozheniya, 5(3), 1-9 (1971).

[2] Harnack, A.: Über Vieltheiligkeit der ebenen algebraischen Curven. Math. Ann., 10, 189-199 (1876).

[3] Kronheimer, P.B., Mrowka T.S.: The genus of embedded surfaces in the projective plane. Math. Research Letters, 1, 797-808 (1994).

[4] Mikhalkin G., Polyak M.: Whitney formula in higher dimensions. J. Diff. Geom. (to appear).

[5] Petrowsky I.G.: On the topology of real plane algebraic curves. Ann. Math., 39, 187-209 (1938).

[6] Polotovskii, G.M.: Catalog of M-decomposing curves of order 6. Doklady Academii Nauk SSSR, ser. mat., 236(3), 548-551 (1977).

[7] Rokhlin, V.A.: Congruences modulo 16 in sixteenth Hilbert problem. Functsional’ni Analiz i ego Prilozheniya, 6(4), 58-64 (1972).

[8] Viro, O.Ya.: The progress in topology of real algebraic varieties over last six years. Uspehi Mat. Nauk, 41(3), 45-67 (1986).

Dept. of Mathematics, University of Toronto, Toronto, Ont. M5S 3G3, Canada Current address: MSRI, 1000 Centennial Dr., Berkeley, CA 94720, USA

 E-mail address: mihalkin@math.toronto.edu, grisha@msri.org