Real inflection points of real linear series on an elliptic curve
RREAL INFLECTION POINTS OF REAL LINEAR SERIES ON ANELLIPTIC CURVE
ETHAN COTTERILL AND CRISTHIAN GARAY L ´OPEZ
Abstract.
Given a real elliptic curve E with non-empty real part and [ D ] ∈ Pic E its g , we study the real inflection points of distinguished subseries ofthe complete real linear series |L R ( kD ) | for k ≥
3. We define key polynomials whose roots index the ( x -coordinates of) inflection points of the linear series,away from the points where E ramifies over P . These fit into a recursivehierarchy, in the same way that division polynomials index torsion points.Our study is motivated by, and complements, an analysis of how inflec-tionary loci vary in the degeneration of real hyperelliptic curves to a metrizedcomplex of curves with elliptic curve components that we carried out in [1]. Introduction
This note is a companion to the papers [3] and [1]. In [3], the second authordescribed the topology of real inflectionary loci of complete linear series on a realelliptic curve E as a function of the topology of the real locus E ( R ), and used thisas the inductive basis for a construction of real canonical curves of genus four in P with many real Weierstrass points. Then in [1], joint with I. Biswas, we useda degeneration borrowed from non-Archimedean geometry to relate two distinctgeneralizations of real inflectionary loci of complete series on E , namely(1) real inflectionary loci of complete series on real hyper elliptic curves X ofgenus g ≥
2; and(2) real inflectionary loci of in complete series on real elliptic curves.Very roughly, generalization (1) may be degenerated to generalization (2); a bitmore precisely, the degeneration outputs a limit linear series on a metrized complexof real curves, whose models are smooth elliptic real curves E i , i = 1 , . . . , g .Even more precisely, the complete real series |L R | studied in [1] are all multi-ples of the g obtained from pulling back a distinguished point ∞ on P via thehyperelliptic structure morphism π : X → P . With respect to a suitable choiceof affine coordinates, we can realize X as (a projective completion of) y = f ( x ),where deg( f ) = 2 g + 1 and the map π is given by ( x, y ) (cid:55)→ x . Assume that L R is represented by the divisor kD on X , where D is the pullback of ∞ on P . Theinteresting cases are associated with choices for which k > g ≥
2, so hereafterwe assume this. There is a distinguished real basis of holomorphic sections for |L R | = |L R ( kD ) | given by(1) F g = { , x, . . . , x k ; y, yx, . . . , yx k − g − } . Date : a r X i v : . [ m a t h . AG ] A p r ETHAN COTTERILL AND CRISTHIAN GARAY L ´OPEZ
The ramification locus R π of π is comprised of real points that degenerate to theramification points of the elliptic components E i , viewed as double covers of P .Furthermore, the 2 k − g + 1 sections F g degenerate to 2 k − g + 1 sections that wewill also abusively denote by F g ; these, in turn, determine an inflectionary basisin every point of the ramification locus of E i → P . Hereafter, we assume i = 1.Abusively, we let π denote the double cover E = E → P . In [1], we explicitlycomputed the contribution to (real) inflection made by each point of R π . Note thatin general the inflection arising from R π is not simple, which contrasts with thebehavior of inflectionary loci of complete series. The behavior of inflectionary loci(including local multiplicities) away from R π , however, was unclear. Here we willaddress this mystery behavior through a more careful analysis of the Wronskianswhose zero loci select for (the x -coordinates of) inflectionary loci.One upshot of our analysis will be that inflection is in fact simple (i.e. of multi-plicity one) away from R π , at least when the topology of E ( R ) is maximally real.One of the key features of the complete series case on which the analysis of [3] ispredicated is that inflection points of a complete linear series on an elliptic curvecorrespond to torsion points. Torsion points, in turn, form a sub-lattice of thelattice Λ for which E ( C ) ∼ = C / Λ as a topological space. Their x -coordinates arecomputed by the division polynomials described in [5]. By analogy, we will define (generalized) key polynomials that compute the x -coordinates of inflection pointsof F g on E away from R π . Our key polynomials are extracted from Wronskians,which also play an important rˆole in Griffiths and Harris’ modern treatment [4] ofPoncelet’s theorem on polygons inscribed (and circumscribed) in conics. However,we organize the information encoded by these determinants in an apparently novelway. When E ( R ) is maximally real, we can realize E = E ( λ ) as an elliptic curvein Legendre form, where λ is a real parameter. The inflectionary loci of curvesin the associated flat family then determine a plane curve C = C ( k, g ), whose sin-gularities control in a rather precise way the (real) inflectionary loci of the curves E ( λ ). Hopefully we will manage to convince the reader that the further study ofkey polynomials and curves is interesting in its own right.The authors would like to acknowledge the valuable help of Eduardo Ruiz-Duarte. 2. Wronskians, revisited
Borrowing notationally from [1], assume that E is defined by an affine equation y = f , where f ∈ R [ x ] is a separable cubic. The restriction of the real inflectionarylocus on E to the affine chart U y where the coordinate y is nonvanishing is the zerolocus of (the restriction of) a regular function α equal to a Wronskian determinantof partial derivatives of sections of F , namely α | U y = det( f ( j ) i ) ≤ i,j ≤ k − g . Here f i refers to the i -th element of the distinguished basis (1) of F g , and thesuperscript denotes the (order of) differentiation with respect to x .Now, for any given n ≥
1, define P n ( x ) ∈ R [ x ] by(2) y ( n ) = f − n yP n ( x ) . EAL INFLECTION POINTS OF REAL LINEAR SERIES ON AN ELLIPTIC CURVE 3
Note that α | U y = y ( n ) precisely when n = k + 1 ≥ g = k −
1. It follows inthis situation that the roots x = γ of P n ( x ) for which f ( γ ) > x -coordinatesof the inflection points of the codimension-( g −
1) subseries of |L R | spanned by F g away from the ramification locus R π . (Note that the positivity of f is required inorder for the root of a generalized key polynomial to lift to an inflection point.)Accordingly, here we will focus on the calculation of the key polynomials P n ( x ),which may be done recursively. Differentiating (2) yields y ( n +1) = dP n dx · f − n y + P n ( − nf − ( n +1) f (cid:48) y + f − n y (cid:48) )= dP n dx · f − n y + P n ( − nf − ( n +1) f (cid:48) y + f − n · f (cid:48) f − y )= f − ( n +1) y · (cid:18) dP n dx · f + P n f (cid:48) ( − n + 1 / (cid:19) from which we deduce that(3) P n +1 = dP n dx · f + ( − n + 1 / P n · dfdx . The key polynomials P n , n ≥
1, are determined by the recursion (3) together withthe seed datum(4) P = 12 dfdx . Empirically we observe the following phenomenon.
Conjecture 2.1.
When the polynomial f has three distinct real roots, the n th keypolynomial P n is separable, for all n ≥ . Indeed, the assumption that f has three distinct real roots means that the topol-ogy of E ( R ) is maximally real, and that E is isomorphic as a real curve to one ofthe Legendre form(5) y = x ( x − x − λ )where λ is a real parameter, λ (cid:54) = 0 ,
1. We have checked that Conjecture 2.1 holdswhenever n ≤ x -discriminant of P n for the ellipticcurve (5) and checking that the only real roots are 0 and 1 (which appear with largemultiplicities). Note that the discriminant of P is a polynomial in λ of degree 218.On the other hand, when f has two conjugate roots, P n is no longer separablein general, whenever n ≥
4. For example, one checks that the discriminant of P ofthe curve with normal form(6) y = x ( x − (1 + λi ))( x − (1 − λi ))has a nontrivial real root when λ = ± √ . So Conjecture 2.1, assuming it holds,strongly depends on the maximal reality of E ( R ).Curiously, it also appears that the degree of the x -discriminant of P n dependsupon the topological type of E ( R ). Conjecture 2.2.
Let f = f ( x, λ ) be a cubic polynomial associated to a one-parameter family of elliptic curves y = f as in (5) or (6) . Define the associatedkey polynomials P n = P n ( x, λ ) according to the recursive hierarchy (3) together ETHAN COTTERILL AND CRISTHIAN GARAY L ´OPEZ with the seed (4) . When f is of Legendre type (5) (resp., of type (6) ), the degree ofthe x -discriminant of P n is equal to n − n + 2 (resp., n − n ). Hereafter, we focus our attention on elliptic curves E = E ( λ ) arising from the realLegendre family (5). We will be concerned with generalized key polynomials whoseroots encode the x -coordinates of inflection points of F g away from the ramificationlocus R π of π : E → P , or equivalently, inflection points whose x -coordinates arenot roots of f . By substituting y ( m ) = f − m yP m in the n = ( k + 1)-th Wronskiandeterminant that defines α | U y , we can write α | U y = ( f − n y ) µ P µ,n , where P µ,n isnow the generalized key polynomial associated with F g when k − g = µ ≥
1. This P µ,n is a polynomial of degree µ in the key polynomials P m = P ,m . We now makethis explicit in the cases µ = 2 , Case: µ = 2 . Here(7) α | U y = det (cid:18) y ( n ) y ( n +1) ( xy ) ( n ) ( xy ) ( n +1) (cid:19) = ( f − n y ) P ,n ;applying the facts that(8) y ( m ) = f − m yP m and ( xy ) ( m ) = my ( m − + xy ( m ) for all m ≥ α | U y = ( n + 1)( y ( n ) ) − ny ( n +1) y ( n − and(9) P ,n = ( n + 1) P n − nP n − P n +1 for all n ≥ Case: µ = 3 . This time, we have(10) α | U y = det y ( n ) y ( n +1) y ( n +2) ( xy ) ( n ) ( xy ) ( n +1) ( xy ) ( n +2) ( x y ) ( n ) ( x y ) ( n +1) ( x y ) ( n +2) = ( f − n y ) P ,n . Evaluating the Wronskian determinant (10) and applying the derivative identities(8) together with the fact that(11) ( x y ) ( m ) = m ( m − y ( m − + 2 mxy ( m − + x y ( m ) whenever m ≥ P ,n = ( n + 1) ( n + 2) P n − n ( n + 1) P n (2( n + 2) P n − P n +1 + ( n − P n − P n +2 )+ n (( n + n − P n − P n +1 + n ( n + 1) P n − P n +2 )(12) for all n ≥
2. 3.
Real roots of generalized key polynomials
In his Ph.D. thesis [3, Prop. 3.2.5], the second author obtained a completecharacterization of the real inflectionary loci of complete real linear series |L R | ofdegree d ≥ E . He showed, in particular, that when the reallocus E ( R ) has two real connected components, the number of real inflection pointsof |L R | is either d or 2 d , depending upon whether d is odd or even. In the presentsetting, d = 2 k is always even, and therefore |L R | has precisely 4 k real inflectionpoints. EAL INFLECTION POINTS OF REAL LINEAR SERIES ON AN ELLIPTIC CURVE 5
On the other hand, the linear series spanned by F g is complete precisely when n = µ + 2, i.e. when k = µ + 1. In that situation, there are 4 µ + 4 real inflectionpoints, of which 4 µ lie away from R π . It follows that the corresponding generalizedkey polynomial P µ,µ +2 = P µ,µ +2 ( λ ) associated with any choice of real parameter λ has precisely 2 µ real roots x = γ for which f ( γ, λ ) > P µ,µ +2 determinesthe real inflectionary behavior of every generalized key polynomial P µ,n = P µ,n ( λ )whenever n ≥ µ + 2, and irrespective of the choice of λ . Namely, graphing the zeroloci ( P µ,n ( x, λ ) = 0) when µ = 1 , , n ≥ µ + 2, we are led tospeculate the following. Conjecture 3.1.
Let µ ≥ and n ≥ µ + 2 be nonnegative integers. For everyfixed value of λ (cid:54) = 0 , , the corresponding generalized key polynomial P µ,n ( λ ) hasprecisely either µ or µ real roots x = γ such that f ( γ, λ ) > , depending uponwhether n − µ is odd or even. Equivalently, the conjecture predicts that the codimension- ( g − linear subseriesof |L R | spanned by F g has precisely either k − g ) or k − g ) real inflection pointsaway from R π , depending upon whether g ≥ is even or odd .4. Real loci of plane curves defined by key polynomials
Degrees and symmetries of key polynomials.
We begin by showing that P n = P n ( x, λ ) is of degree exactly 2 n in x (a consequence of Pl¨ucker’s formula for n ≥
4) and of degree exactly n in λ . Lemma 4.1.
For every positive integer n ≥ , we have deg x ( P n ) = 2 n and deg λ ( P n ) = n. Proof.
The claim is clear for n = 1 ,
2; so we assume that n ≥
3. First recall thatsince k = g + 1, we have n = k + 1 = g + 2. So the geometric object of interest is a g g +32 g +2 on the elliptic curve E , whose total inflectionary degree may be realized as(13) (2 g + 2)( g + 3) = I R π + 2deg x ( P g +2 )where I R π is the inflection concentrated in the ramification points, which is 4 (cid:0) g +12 (cid:1) +2( g − x ( P g +2 ) = 2 g + 4 = 2( g + 2), as desired.To prove the second statement, we argue by induction. The claim is obviouswhen n = 1; so assume that it holds for all k ≤ n for some n >
1. We then have P n = a n ( x ) λ n + Q n ( x, λ )with a n ( x ) ∈ R [ x ] and deg λ ( Q n ) < n . Applying (3) to P n writen in this form, wesee that the coefficient of λ n +1 in P n +1 is the polynomial a n +1 ( x ) = − x ( x − da n ( x ) dx + ( − n + 1 / − x ) a n ( x ) . If a n +1 ( x ) = 0, then a n ( x ) is a solution of the following differential equation y (cid:48) y = R ( x ) = ( − n + 1 / − x ) x ( x − y = Ke (cid:82) R ( x ) dx = K ( x − x ) (2 n − / is not a polynomial, so a n +1 ( x ) (cid:54) = 0 anddeg λ ( P n +1 ) = n + 1. (cid:3) ETHAN COTTERILL AND CRISTHIAN GARAY L ´OPEZ
We next show that the curve defined by P n = 0 has special symmetry. Lemma 4.2.
For every positive integer n ≥ , we have (14) P n ( x, λ ) = P n ( x, z ) and P n ( x + 1 , λ + 1) = P n ( − x, − λ ) . Here by P n ( x, z ) we mean the polynomial of degree n obtained by homogenizing withrespect to z to obtain a degree- n polynomial in R [ x, λ, z ] ; and then dehomogenizingwith respect to λ to obtain a degree- n polynomial in R [ x, z ] .Proof. We prove (14) by applying the basic recursion (3). Clearly both properties(14) hold when n = 1. For the first property, note that f ( x, λ ) = f ( x, z ); the factthat the analogous property holds for P n is now immediate by induction using (3).We now focus on the on the second symmetry in (14). By induction, we may assumeit holds for all n ≤ N , for some N ≥
1. Then because (the second symmetry in) (14)is preserved under taking products and multiplying by scalars, ( − n + ) P n · dfdx alsosatisfies (14). It is slightly more delicate, but nevertheless true, that dP n dx · f satisfies(14). Indeed, one checks easily that f ( x + 1 , λ + 1) = − f ( − x, − λ ), so it sufficesto check that the x -derivative of P n satisfies the same antisymmetric property, butthis follows easily from the chain rule. (cid:3) An upshot of Lemma 4.2 is that the monodromy group associated to the projec-tion from the point [0 : 0 : 1] of the projective closure C ( n ) of the curve C ( n ) definedby P n = 0 inside of P x,λ,z contains transpositions that freely permute the points p , p , and p with coordinates [0 : 0 : 1], [0 : 1 : 0], and [1 : 1 : 1]. In particular,the singularities of C ( n ) in these three points are analytically isomorphic. Conjecture 4.3.
For every positive integer n ≥ , the plane curve C ( n ) is nonsin-gular along P \ { p , p , p } . Note that Conjecture 4.3 is easy to verify by computer (i.e. algorithmically,using Groebner bases) whenever n is small. However, we have thus far been unableto show “by hand” that the system of equations P n ( x, λ ) = dP n dx = dP n dλ is exactlysolved by the coordinate pairs (0 ,
0) and (1 ,
1) in general. The next four subsectionsare devoted to proving Conjecture 3.1 in the first nontrivial cases.
Theorem 4.4.
Conjecture 3.1 holds when µ = 1 and n = 2 , , , . In fact, the case n = 2 is non-geometric (since k = 1 in that case) and when n = 3 the associated linear series is complete (so in particular Conjecture 4.3 holdson the basis of [3, Thm 3.2.5]), so the first geometrically interesting case n = 4.However as the argument used in the proof when n = 4 is a more elaborate versionof the argument for n = 3 (which is itself a more elaborate version of that usedfor n = 2) we find it instructive to include a discussion of the n = 2 and n = 3cases as well. The key point here is that the recursion (3) implies that the n thkey polynomial P n = P n ( x, λ ), which is of degree 2 n in x , is only of degree n in λ .Accordingly we can obtain a global parametrization λ = λ ( x ) for λ in terms of x whenever n ≤ Case: n = 2 . We begin by writing P as a quadratic polynomial in λ withcoefficients in R [ x ], namely P ( x, λ ) = − λ + (cid:18) − x + 32 x (cid:19) λ + (cid:18) − x + 34 x (cid:19) . EAL INFLECTION POINTS OF REAL LINEAR SERIES ON AN ELLIPTIC CURVE 7
Applying the quadratic formula, we deduce that λ = λ ( x ) = ( − x + 3 x ) ∓ x ( x − / solves P = 0. Conjecture 3.1 in this particular case follows easily.4.3. Case: n = 3 . We have P ( x, λ ) = (cid:18) − x + 38 (cid:19) λ + (cid:18) x − x (cid:19) λ + (cid:18) x − x (cid:19) λ + (cid:18) − x + 34 x (cid:19) . Whenever x (cid:54) = , the leading coefficient − x + of P is nonzero. Assuming thisto be the case, we may divide by the leading coefficient. Doing so produces a moniccubic (cid:101) P in λ whose coefficients are (formally) power series in x , namely (cid:101) P ( x, λ ) = λ + (cid:18) −
52 + 12 α (cid:19) xλ − (1 + 4 α ) x λ + (cid:18)
12 + 32 α (cid:19) x where α = − x . Now set a i := [ λ i ] (cid:101) P , i = 0 , ,
2. Changing variables according to λ (cid:55)→ λ ∗ where λ ∗ = λ + a λ + (cid:18) −
56 + 16 α (cid:19) x allows (cid:101) P to be rewritten as a depressed cubic , namely (cid:101) P ∗ = (cid:101) P ∗ ( x, λ ∗ ) = ( λ ∗ ) + pλ ∗ + q where p = a − a − (1 + 4 α ) x + (cid:18) − α − α (cid:19) x and q = a − a a a
27 = 13 (2 α − α − x + 1108 ( α − x . Cardano’s formula now establishes that λ ∗ = u + v solves (cid:101) P ∗ = 0, where(15) u = (cid:118)(cid:117)(cid:117)(cid:116) − q (cid:115)(cid:18) q (cid:19) + (cid:18) p (cid:19) and v = (cid:118)(cid:117)(cid:117)(cid:116) − q − (cid:115)(cid:18) q (cid:19) + (cid:18) p (cid:19) . Note that when ∆ = ∆( x ) := (cid:18) q (cid:19) + (cid:18) p (cid:19) is negative, the expression for u in (15) selects for an arbitrary cube root, and v := u . In the latter situation, the corresponding cubic (cid:101) P ∗ = (cid:101) P ∗ ( x ) has three realroots.In our case, we have ∆ = x α R , where R = − x − . In particular, ∆ is negative whenever x / ∈ { , , } . From (15) we deduce that(16) u = x ( x − x − · (cid:115) (cid:18) x − (cid:19) + 5216 + 13 √ x ( x − (cid:18) x − (cid:19) √− . Further, by negativity of the discriminant ∆, we may always distinguish threedistinct roots of (cid:101) P ∗ ( x ) given by u + u , ζu + ζu , and ζ u + ζ u , where ζ is a ETHAN COTTERILL AND CRISTHIAN GARAY L ´OPEZ primitive cube root of unity and u is as in (16), assuming that an arbitrary cubicroot of − q/ √ ∆ is fixed.Now let (cid:101) v = (cid:101) v ( x ) := 118 (cid:18) x − (cid:19) + 5216 + 13 √ x ( x − (cid:18) x − (cid:19) √− . We can rewrite (cid:101) v as cos( γ ) + sin( γ ) √−
1, where γ = γ ( x ∗ ) := arctan (cid:32) √ · x ∗ (( x ∗ ) − )( x ∗ ) + (cid:33) and where x ∗ = x − . Putting everything together, we obtain three distinguishedsolutions (cid:101) λ i = (cid:101) λ i ( x ∗ ), i = 1 , , P = P ( x ∗ ) = 0, given by (cid:101) λ = (cid:18) ( x ∗ ) − x ∗ (cid:19) · (cid:18) arctan (cid:18) √ · x ∗ (( x ∗ ) − )( x ∗ ) + (cid:19)(cid:30) (cid:19) + 56 x ∗ + 12 + 124 · x ∗ , (cid:101) λ = (cid:18) ( x ∗ ) − x ∗ (cid:19) · (cid:18) arctan (cid:18) √ · x ∗ (( x ∗ ) − )( x ∗ ) + (cid:19)(cid:30) π/ (cid:19) + 56 x ∗ + 12 + 124 · x ∗ , and (cid:101) λ = (cid:18) ( x ∗ ) − x ∗ (cid:19) · (cid:18) arctan (cid:18) √ · x ∗ (( x ∗ ) − )( x ∗ ) + (cid:19)(cid:30) π/ (cid:19) + 56 x ∗ + 12 + 124 · x ∗ . Conjecture 3.1 now follows in the present case from the following subsidiaryclaims about the real locus of the affine plane curve C (3).(1) When x <
0, the real locus is comprised of three infinite x -monotone (i.e.nondecreasing with respect to x ) curves λ i = λ i ( x ), i = 1 , , λ > > λ > x > λ . These curves intersect in the limit point (0 ,
0) (whoseassociated value of λ describes a singular rational curve). Moreover, λ hasa vertical asymptote at x = ; and −∞ = lim x →−∞ λ = lim x →−∞ λ = − lim x →−∞ λ . (2) When 0 < x <
1, the real locus is a union of monotone curves λ i , 4 ≤ i ≤ λ > x > λ > λ ; λ , λ , and λ intersect in (0 , λ and λ connect(0 ,
0) with (1 , λ . Moreover, we have λ > λ and lim x → − λ = − lim x →
12 + λ = −∞ . (3) When x >
1, the real locus is comprised of three infinite monotone curves λ i , i = 8 , ,
10, where λ > x > λ > λ . These intersect in their commonlimit point (1 , E ( λ ) is singular. Moreover,we have ∞ = lim x →∞ λ = lim x →∞ λ = − lim x →∞ λ (cid:3) Case: n = 4 . Here our analysis closely follows that of the n = 3 case. Tobegin, we have P ( x, λ ) = (cid:18) − x +3 x − (cid:19) λ + (cid:18) x − x +3 x (cid:19) λ + (cid:18) − x + 212 x − x (cid:19) λ + (cid:18) − x + 214 x (cid:19) λ + (cid:18) x − x (cid:19) . EAL INFLECTION POINTS OF REAL LINEAR SERIES ON AN ELLIPTIC CURVE 9
Now set β = x − x + = x − ) + . Note that β is nonzero for all real values of x . Dividing P by its leading coefficient yields (cid:101) P ( x, λ ) = λ + (cid:18) − x −
14 + (cid:18) − x + 564 (cid:19) β (cid:19) λ + (cid:18) x + 78 x + 65128 + (cid:18) x − (cid:19) β (cid:19) λ + (cid:18) x − x − x − x − x − (cid:18) − x + 132516384 (cid:19) β (cid:19) λ + (cid:18) − x + 516 x + 95256 x + 35128 x + 6454096 x + 2954096 x + 149565536 + (cid:18) x − (cid:19) β (cid:19) . Let b i := [ λ i ] (cid:101) P ( x, λ ), 0 ≤ i ≤
3. Changing variables according to λ (cid:55)→ λ ∗ with λ = λ ∗ − b enables us to convert (cid:101) P ( x, λ ) to a depressed quartic (cid:101) P ∗ ( x, λ ∗ ) of theform ( λ ∗ ) + p ( λ ∗ ) + qλ ∗ + r , where p = b − b , q = 18 b − b b + b , and r = − b + b − b b + 116 b b . According to Ferrari, the roots of (cid:101) P ∗ ( x, λ ∗ ) are given by(17) λ ∗ = ± √ m ± (cid:113) − (2 p + 2 m ± √ q √ m )2where m denotes any root of the resolvent cubic K ( z ) = 8 z + 8 pz + (2 p − r ) z − q associated with (cid:101) P ∗ . Here the monic depressed cubic associated with K is K = z − (cid:18) p + r (cid:19) z + (cid:18) − p + 13 rp − q (cid:19) = z + (cid:18) − b − b + 14 b b (cid:19) z + (cid:18) − b + 13 b b − b + 124 b b b − b b (cid:19) . Now let p := [ z ] K and q := [ z ] K . From Cardano it follows that roots of K are given by m = u + v , where u = (cid:114) − q (cid:112) ∆ and v = (cid:114) − q − (cid:112) ∆ . Here ∆ := ( q ) + ( p ) is the discriminant of K . In this case q = − β x ( x − (cid:18)(cid:18) x − (cid:19) + 56 (cid:18) x − (cid:19) + 11432 (cid:19) and ∆ = β x ( x − (cid:18)(cid:18) x − (cid:19) + 5108 (cid:19) . It follows that u = β x ( x − (cid:112) U + √ U and v = β x ( x − (cid:112) U − √ U where U := (cid:18) x − (cid:19) + 56 (cid:18) x − (cid:19) + 11432 and U := x ( x − (cid:18)(cid:18) x − (cid:19) + 5108 (cid:19) . Here U is positive for all real values of x , while U is positive (resp., negative)whenever x < x > < x < U − U = 6427 (cid:18)(cid:18) x − (cid:19) + 112 (cid:19) which is clearly positive for all real-valued x . Accordingly, whenever x < x > (cid:112) U ( x ) to mean the positive square root of U ( x ); it follows that U − √ U and U + √ U are positive. In this situation, we denote by u = u ( x ) and v = v ( x )the real functions that select for β x ( x − times the unique real positive cubic roots of U + √ U and U − √ U , respectively. Similarly, whenever 0 < x < (cid:112) U ( x ) to mean √− − U ( x ), and then U + √ U and U − √ U become complex conjugates of one another, with positivereal parts. Finally, we obtain m = m − p = β x ( x − (cid:113) U + √ U + β x ( x − (cid:113) U − √ U − b + 18 b = β x ( x − (cid:18) γ + 7 β (cid:18)(cid:18) x − (cid:19) + 128 (cid:19)(cid:19) . (19) where γ = γ ( x ) := (cid:112) U + √ U + (cid:112) U − √ U . Note here that γ satisfies thealgebraic equation γ − U − x − ) + ) γ = 0 . It follows from the above discussion that our choice of m in (19) is always positive,and it remains to control for the sign of the expression inside the second radical in(17). Here p + m = m + 23 p = β x ( x − (cid:18) γ − β (cid:18)(cid:18) x − (cid:19) + 128 (cid:19)(cid:19) while q = β (cid:18) x − (cid:19) x ( x − (cid:18)(cid:18) x − (cid:19) + 732 (cid:18) x − (cid:19) + 1128 (cid:19) . We claim that(i) − ( √ m ( p + m ) − q ) ≥ ⇐⇒ x ≥
1, and(ii) − ( √ m ( p + m ) + q ) ≥ ⇐⇒ x ≤ P n and their derivativesfor small values of n (namely, the validity of Conjecture 4.3) it suffices to check thatthe signs of − ( √ m ( p + m ) ± q ) are as claimed locally near x = 0 and x = 1. Thismay be achieved by computing the values of √ m ( p + m ) ± q and its first derivativein x = 0 and x = 1. We omit the explicit calculation.Finally, the explicit “parametrization” (17) for C (4) may be used to check thatthe real locus C (4)( R ) has the following features, and the validity of Conjecture 3.1when n = 4 follows easily.(1) There are no points in the real locus for which 0 < x < x <
0, there are two monotone curves λ i = λ i ( x ), i = 1 , λ > > λ > x . These curves intersect in their common limit of (0 , x →∞ λ = ∞ = − lim x →∞ λ . (3) When x >
1, there are two monotone curves λ i = λ i ( x ), i = 3 , x > λ ( x ) > λ ( x ). These curves intersect in their common limit of (1 , x →∞ λ = ∞ = − lim x →∞ λ . EAL INFLECTION POINTS OF REAL LINEAR SERIES ON AN ELLIPTIC CURVE 11
Case: n = 5 . This case is more difficult than the preceding ones. Indeed, asnoted in Section 4, the monodromy group G of P ( x, λ ) over C ( x ) associated tothe projection of C (5) from (0 , ,
1) contains an element of order 3. On the otherhand, the solvable subgroups of S are precisely the subgroups of the Frobeniusgroup F of order 20. It follows that G is not solvable, and therefore no globalsolution λ = λ ( x ) of the plane curve with equation P ( x, λ ) = 0 is available. Onthe other hand, local (Puiseux) parametrizations λ = λ ( x ) that solve P n ( x, λ ) = 0are always available (see, e.g. [2, Sec. 8.3]), and we will carry this out now when n = 5 near the singular point p ; in view of (4.2), the local descriptions of C (5)near the singular points p , p , and p are identical. Local parametrizations at (0 , . We have P ( x, λ ) = − x + 7516 x − x λ + 7516 x λ − x λ + 1352 x λ − x λ + 157516 x λ + 452 x λ − x λ + 256516 x λ − x λ − xλ + 193532 x λ − x λ + 1352 x λ + 10532 λ − xλ + 452 x λ − x λ . (20) From (20) it follows that the lower hull of the Newton polygon consists of the linesegments (cid:96) , (cid:96) between (0 ,
5) and (3 ,
2) (of slope − ,
2) and (9 ,
0) (of slope − ), respectively. So there are at least two local (complex) branches near (0 , (cid:96) i . Branches associated with (cid:96) . Begin by writing(21) λ = x ( c + λ )where c ∈ C ∗ and λ = λ ( x ) = c x γ + c x γ + γ + . . . is a Puiseux series that remains to be determined. Now substitute the first approx-imation (21) into the implicit equation P = 0 and divide by the smallest commonpower in x ; the result is a polynomial in x , c , and λ the sum of whose lowest x -order terms is1532 x c (7 c − c + 48 c −
32) = 1532 x c ( c − c − c + 16) . As the lowest x -order terms of P must sum to zero, we deduce that either c = 2or else c is one of the two complex conjugate roots of 7 c − c + 16. In the lattercase, we obtain two non-real branches, exchanged under conjugation. So say that c = 2. Then upon dividing by x we obtain P ( x, λ ) = 16 x − x + 104 x − x + 17 x + 48 λ − xλ + 280 x λ − x λ − x λ + 10 x λ + 96 λ − xλ + 648 x λ − x λ + 88 λ − xλ + 598 x λ − x λ + 40 λ − xλ + 272 x λ − x λ + 7 λ − xλ + 48 x λ − x λ . (22) The lower hull of the Newton polygon of P ( x, λ ) is a single line segment ofslope − ,
1) and (1 , γ i = 1 for all i ≥ λ = (cid:80) ∞ i =1 c i +1 x i in (22) and setting the result equal to zero, we maycompute all of the remaining power series coefficients c i , i ≥
2. Clearly the branchin question is real.
Branches associated with (cid:96) . Consider now those branches arising from the linesegment (cid:96) . Much as before, begin by writing(23) λ = x ( c + λ ) where λ = λ ( x ) = c x γ + c x γ + γ + . . . is a Puiseux series that remains to bedetermined. Substituting (23) into the implicit equation P = 0 and dividing bythe smallest common power in x yields a polynomial in x , c , and λ whose sum oflowest x -order terms is 1532 x (10 − c ) . It follows that c = ± √ . Now say that c = √ . Then dividing by − x weobtain a polynomial P ( x, λ ) = − x + 80640 x + 7680 √ x − x + 14080 √ x + 3000 x − √ x − x + 20160 √ x + 20800 x − √ x − x + 750 √ x − √ x + 800 √ x + 16384 √ λ − √ xλ + 129024 √ x λ + 189440 x λ − √ x λ − x λ + 9600 √ x λ + 241920 x λ − √ x λ − x λ + 66560 √ x λ + 15000 x λ − √ x λ − x λ + 16000 x λ + 32768 λ − xλ + 258048 x λ − √ x λ − x λ + 159744 √ x λ + 57600 x λ − √ x λ − x λ + 193536 √ x λ + 399360 x λ − √ x λ − x λ + 24000 √ x λ − √ x λ + 25600 √ x λ − x λ + 212992 x λ − x λ + 30720 √ x λ + 258048 x λ − √ x λ − x λ + 212992 √ x λ + 96000 x λ − √ x λ − x λ + 102400 x λ + 30720 x λ − x λ + 212992 x λ − √ x λ − x λ + 38400 √ x λ − √ x λ + 40960 √ x λ − x λ + 30720 x λ − x λ + 32768 x λ whose Newton polygon’s lower hull is a single line segment of slope − ,
1) and (1 , γ i = 1 for all i ≥
2, and thatall remaining power series coefficients c i , i ≥ λ = (cid:80) ∞ i =1 c i +1 x i in P ( x, λ ) = 0. The corresponding branch is real.Finally, when c = − √ we obtain a polynomial P ( x, λ ) = P ( x, − λ ). Thecorresponding branch is clearly real as well.To conclude, it now suffices to show that C (5)( R ) has the same characteristicfeatures as in the n = 3 case; these collectively determine a topological profile forthe real locus, and we conjecture that the following is true. Conjecture 4.5.
Let n ≥ be a positive integer. Then C ( n )( R ) and C ( n mod 2 +2)( R ) have the same topological profile. Clearly Conjecture 4.5 implies Conjecture 3.1. For the sake of completeness, wealso record the following conjecture, which is related to Conjectures 4.5 and 4.3.
Conjecture 4.6.
For all n ≥ , the set of real roots of the discriminant ∆ n =∆ n ( x ) of P n = P n ( x, λ ) with respect to λ is { , } . Figure 1 includes computer-generated sketches of the real loci of C ( n ) for someeven values of n ; the graphs give compelling evidence for Conjectures 4.5 and 3.1.The affirmation of Conjecture 4.6 is a consequence by the monotonicity propertyof Conjecture 4.5, together with the conjectural characterization 4.3 of the realsingular points of C ( n ). In particular, Conjecture 4.6 holds when n = 5.It remains to verify Conjecture 4.5 when n = 5. To this end, first note thatimplicitly differentiating the equation P n ( x, λ ( x )) = 0 with respect to x shows that dλdx = 0 ⇐⇒ dP n dx = 0 EAL INFLECTION POINTS OF REAL LINEAR SERIES ON AN ELLIPTIC CURVE 13 - - - - - - - - - - - - - - - - a ) b ) c ) d ) Figure 1.
Real loci of a) P = 0, b) P = 0, c) P = 0, and d) P = 0.away from the singular points of C ( n ). On the other hand, for small values of n (including n = 5), the equations P n = dP n dx = 0 have no common affine solutions( x, y ) in R \ { (0 , , (1 , } . In light of the Puiseux parametrizations of the realbranches near (0 ,
0) and the symmetries (4.2), the desired x -monotonicity propertiesof the solution curves λ = λ ( x ) follow immediately. It suffices now to check that P ( x, λ ) = 0 has exactly three solutions λ for each fixed value of x in R \ { , , } ,i.e. that C (5)( R ) is connected over each of the critical intervals ( −∞ , , , ∞ ) in x . But this, in turn, is an immediate consequence of the Implicit Functiontheorem.Finally, it remains to verify the inequalities of the real solution curves λ = λ ( x )described in the topological profile of C (3) above relative to x . These hold locallynear the singular points (0 ,
0) and (1 ,
1) of C (5) in the affine plane, so it sufficessimply to show that the intersection of the line y = x with C (5)( R ) is precisely supported along { (0 , , (1 , } . And indeed, we have P ( x, x ) = − x + 52532 x − x + 52516 x − x + 10532 x = − x ( x − . (cid:3) Figure 2 includes sketches of the the real loci of C ( n ) for small odd values of n ; just as in Figure 1, the validity of Conjectures 4.5 and 3.1 in these cases isgraphically apparent. a ) b ) c ) d ) Figure 2.
Real loci of a) P = 0, b) P = 0, c) P = 0, and d) P = 0.5. From incomplete series on an elliptic curve to complete series ona hyperelliptic curve
Conjectures 4.5 and 3.1, assuming they hold, may be leveraged to constructcomplete real linear series on real hyperelliptic curves of genus g ≥ EAL INFLECTION POINTS OF REAL LINEAR SERIES ON AN ELLIPTIC CURVE 15
Theorem 5.1.
Assume that Conjecture 3.1 holds. There exists a maximally-realhyperelliptic curve X with affine equation y = f admits a complete real linearseries |L R | = |L R ( kD ) | with the distinguished basis F g as in (1) whose total realinflectionary degree is ω R ( k, g ) = 2 g ( g + 1) + 2( k − g )( g −
1) + 2 g (1 + g mod 2)( k − g ) . Here f = f ( x ) is a polynomial of degree g − ramified over ∞ ∈ P , and D is thedivisor represented by the pullback of ∞ to X .Proof. Follows immediately from Theorems 5.8 and 6.1 of [1]. (cid:3)
References [1] I. Biswas, E. Cotterill, and C. Garay L´opez,
Real inflection points of real hyperelliptic curves , arXiv:1708.08400 .[2] E. Brieskorn and H. Kn¨orrer, “Plane algebraic curves”, Springer, 1986.[3] C. Garay, Tropical intersection theory, and real inflection points of real algebraic curves ,Univ. Paris VI doctoral thesis, 2015.[4] P. Griffiths and J. Harris,
On Cayley’s explicit solution to Poncelet’s porism , Enseign. Math. (1978), 31–40.[5] J. Silverman, “The arithmetic of elliptic curves”, Springer, 1985. Instituto de Matem´atica, UFF, Rua Prof Waldemar de Freitas, S/N, 24.210-201Niter´oi RJ, Brazil
E-mail address : [email protected] Departamento de Matem´aticas, Centro de Investigaci´on y de Estudios Avanzados delIPN, Apartado Postal 14-740, 07000. Ciudad de M´exico, M´exico.
E-mail address ::