Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves

We show that for a class of two-loop diagrams, the on-shell part of the integration-by-parts (IBP) relations correspond to exact meromorphic one-forms on algebraic curves. Since it is easy to find such exact meromorphic one-forms from algebraic geometry, this idea provides a new highly efficient algorithm for integral reduction. We demonstrate the power of this method via several complicated two-loop diagrams with internal massive legs. No explicit elliptic or hyperelliptic integral computation is needed for our method.


Introduction
With the beginning of Run II of the Large Hadron Collider (LHC), we need high precision scattering amplitudes in Quantum Chromodynamics and the Standard Model, to reduce the theoretical uncertainty. The precise scattering amplitude computation suffers from problems of the large number of loop Feynman diagrams and difficult integrations for each loop diagram. This paper aims at developing a new method of reducing loop integrals to the minimal set of integrals, i.e., master integrals (MIs).
Traditionally, integral reduction can be achieved by applying integration-by-parts (IBP) identities [1] and considering the other symmetries of loop diagrams. However, given a twoloop or higher-loop integral, it is difficult to find a particular IBP identity which reduce it to MIs without introducing unwanted terms. There are several implements of IBPs generating codes AIR [2], FIRE [3][4][5] and Reduze [6,7], based on Laporta algorithm [8], by the computation of Gaussian elimination or Gröbner basis. For multi-loop diagrams with high multiplicities or many mass scales, it may take a lot of time and computer RAM to finish the integral reduction. There are also several new approaches for integral reduction, based on the study of the Lie algebra structure of IBPs [9], Syzygy computation [10,11], reductions over finite fields [12], and differential geometry [13]. Besides, the number of master integrals can be determined by the critical points of polynomials [14].
Schematically, an IBP relation, on the unitarity cut V : D 1 = . . . = D k = 0 becomes contour integral relations [25][26][27][28][29][30][31][32], where the integrals are along combinations of non-trivial cycles of V , contours surrounding singular points of V and poles of ω. If V is an algebraic curve, then the contours are one-dimensional. Furthermore, if V is irreducible, then V has a complex structure and ω is a meromorphic 1-form [33]. In this case, (1.2) holds for all contours, so it implies the ω is an exact form, where F is a meromorphic function on V . So the on-shell part of IBP relations for this diagram correspond to exact meromorphic 1-forms. Mathematically, it is very easy to find exact meromorphic 1-forms on an algebraic curve, so we can quickly get the on-shell IBPs for this diagram.
In this paper, we show several two-loop examples with internal masses for our new method. Multi-loop integrals with internal masses appear frequently in QCD/SM scattering amplitudes, and are bottlenecks for integral reduction or evaluation. We use these complicated cases to show the power of our method: 1. D = 4 planar double box with internal massive legs. The unitarity cut for this diagram is an elliptic curve. The maximal unitarity structure of the symmetric double box, with internal massive legs, was studied in [31]. Here we derive the integral reduction for the general cases, based on the analysis of differential forms on an elliptic curve. We also reduce integrals with doubled propagators based on algebraic curves, which were not considered in [31].
In these examples, we get all the on-shell IBPs analytically. The algorithm is realized by a Mathematica code containing algebraic geometry tools. For each diagram, the analytic integral reduction is extremely fast, which has the time order of minutes. We have the following remarks, • Although the mathematic objects are elliptic or hyperelliptic, we do not need the explicit form of elliptic/hyperelliptic functions, or elliptic/hyperelliptic integrals. Only the differential relations for elliptic and hyperelliptic functions are needed. These relations involve rational coefficients only and are easy to find.
• The method presented in this paper is different from the maximal unitarity method. For the maximal unitarity method, we need to perform contour integrals to extract the master integral coefficients. Our method use the integrand reduction [36][37][38][39] via Gröbner basis [39][40][41][42][43][44][45] first, to reduce the loop amplitude to an integrand basis. Then we use the knowledge of algebraic curves, to reduce the integrand basis further to master integrals. In this way, we avoid the explicit elliptic or hyperelliptic integral computations.
This paper is organized as follows: In section 2, we present our method based on algebraic curves. In section 3 and 4, the double box diagram (elliptic) and sunset diagram (elliptic) with internal masses will be explicitly presented. In section 5, we consider the integral reduction for the massive nonplanar box diagram (hyperelliptic). The rudiments of the knowledge of algebraic curves are included in the appendix.

Integral Reduction via the Analysis of Algebraic Curves
Generically, for a quantum field theory, the L-loop amplitude can be written as [15,16], The set {I k } is called the master integral (MI) basis whose elements are independent loop integrals. In practice, for amplitudes with multiple loops, high multiplicities or several mass scales, it is quite difficult to determine the set of master integral or reduce a generic integral, to the linear combination of master integrals. Traditionally, the integral reduction is done by using IBP identities [1], if there is no boundary term. In general, it is difficult to find the IBP relations for a multi-loop integral reduction.
We present a new way of integral reduction, based on maximal unitarity method and algebraic curves. Given a Feynman integral with k propagators, maximal unitarity method split (2.1) as [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] , Int = i c i I i + integrals with fewer-than-k propagators + rational terms (2.4) where the first sum is over the master integral with exact k propagators. The condition the all internal legs are on-shell, is called the maximal unitarity cut, 5) and the solution set for this equation system is an algebraic variety V . V can be a set of discrete points, algebraic curves or surfaces. (See [46,47] for the detailed mathematical study of multi-loop unitarity cut solutions.) Maximal unitarity replaces the original integral with contour integrals [25][26][27][28][29][30][31][32], schematically, where ω is a differential form on V , and contours c j 's are around the poles of ω and also the nontrivial cycles of V [29,31]. w j are weights of these contours. In particular, to extract the coefficients c i in (2.1), we can find a special set of weights w {i} j [25][26][27][28][29][30][31] such that, For example, the 4D two-loop massless double box diagram contains 7 propagators. The maximal cut is a reducible variety with 6 components [25], each of which is a Riemann sphere. The contours are around the intersecting points of these Riemann spheres, which are singular points of this variety, and also around the poles of ω. For the 4D two-loop double box with massive internal legs, the maximal cut gives one irreducible variety, which is an elliptic curve [31]. There is no singular point on this variety, so the contours are around the poles of ω and also the two fundamental cycles on the elliptic curve. Our observation is that if a differential form ω on V is integrated to zero, around all singular points on V , poles of ω and non-trivial cycles of V .
then from (2.7) and (2.4), the integral corresponding to ω can be reduced to integrals with fewer propagators. Since from the knowledge of algebraic geometry, it is easy to find such ω's satisfying (2.8), we propose a new multi-loop integral reduction method from this viewpoint.
In this paper, we focus on the cases for which the number of propagators equals DL − 1 and the maximal unitarity cut gives one irreducible variety. In such a case, the cut solution V is a smooth algebraic curve with well defined complex structure. The condition (2.8) implies that ω is an exact meromorphic form on V , since the integral is independent of the path and dF = ω. Then from the study of meromorphic functions on V , which is a well-known branch of algebraic geometry, we can list generators for F and then derive all forms which satisfy (2.8).
Explicitly, for this class of diagram, we found that the scalar integral on the cut becomes a holomorphic form on V .
where the 1-form Ω is globally holomorphic (without poles) on V . On the cut, the components of l i 's become meromorphic functions. We can show that these functions generate all meromorphic functions on V . Let F (l 1 , . . . l L ) be a polynomial in the components of loop momenta, then take the derivative of F , The resulting f Ω is an exact meromorphic 1-form. From the analysis above, we get that, so we obtain an integral reduction relation. For the explicit examples in this paper, we can show that this method provides all the on-shell part of integral reduction relations. In practice, our algorithm can be presented as, 1. Use integrand reduction method via Gröbner basis [40,41] to rewrite the loop scattering amplitude as the form of integrand basis. The coefficients of integrand basis can be determined by fusing tree amplitudes or polynomial division of the Feynman integrand.
2. Calculate the maximal cut of the scalar integral to determine the form of holomorphic form Ω as (2.10).
3. Calculate the exterior derivatives of all polynomial F 's. In practice, it is sufficient to consider the linear F 's and then use the chain rule. Then we get all the on-shell IBP relations as (2.11).
4. Use the obtained integral reduction relations to reduce the integrand basis to master integrals.
If the integrand contains doubled-propagator integrals, the algorithm will be slightly different. We need to solve a polynomial Diophantine equation first, and the procedure will be shown in the next section.
Note that our algorithm is different from the traditional maximal unitarity. Usually, maximal unitarity method needs the explicit contour integration to extract the master integral coefficients. However, our algorithm does not require the explicit contour integration, and the explicit form of elliptic/hyperelliptic integral is not needed. The residue computations to find the holomorphic form Ω and the derivative computations (2.11) are much simpler than the contour integrals.

Elliptic Example: Double Box with Internal Masses
The method explained in the previous section can be used for integral reduction for various topologies, for instance, the double-box ( Fig. 1) with three different masses for the internal propagators. The maximal unitarity of the massless double box was discussed in [25]. Then, maximal unitarity structures for double box with 1 ∼ 4 massive external legs were studied in [26,28]. In these cases, the unitarity cuts provide reducible curves.
On the other hand, the unitarity cut of double box with six massive external legs [29] or all massive internal legs provides irreducible curves. The integral reduction for symmetric double box diagram with internal masses, was discussed in [31], via maximal unitarity and the analysis of elliptic functions. Here we show the integral reduction for more generic double box diagram with 3 internal mass scales, based on our new method, without using elliptic functions/integrals explicitly.

Maximal unitarity
The denominators for double box diagrams are We parametrize the loop momenta as, and the Jacobian for this parameterization is, The solutions for the maximal unitarity cut, defines an elliptic curve. To see this, we first solve for the variables α 1 , α 2 , α 3 , β 1 , β 2 and β 3 in terms of α 4 and β 4 , Then the remaining one equation relates α 4 and β 4 , Here A(α 4 ), B(α 4 ) and C(α 4 ) are quadratic polynomials of α 4 , whose coefficients depend on kinematic variables. Formally, β 4 depends on α 4 as, where ∆ is a quartic polynomial in α 4 with four distinct roots, for generic kinematics with internal masses. Hence the maximal unitarity cut defines an elliptic curve, i.e., algebraic curve with genus one, (See the appendix for the basis introduction to elliptic curves.) The choice of keeping α 4 and β 4 and eliminating other variables is purely arbitrary.
On the cut, by a short calculation, the scalar double box integral, has the following structure: where the overall factor is not important for the following discussion.
As [31], it is remarkable that dα 4 √ ∆ is the only holomorphic one-form associated with the elliptic curve. On the cut, the loop-momentum components α i , β i become elliptic functions. So we may study the explicit form of these functions, calculate the elliptic integrals and perform the integral reduction, in a procedure of [31]. However, in this paper, we propose a different procedure: (1) reduce the integrand based on Gröbner basis [40,41] (2) reduce the integrand basis to master integrals by the study of differential forms on the elliptic curve. The advantage of this approach is that the whole computation is very simple: no explicit elliptic parameterization or elliptic integral is needed. The new process can also be easily automated on computer algebra systems.

Integral reduction
We now focus on the double box integral with numerator N , Integrand reduction method via Gröbner basis method [40,41] determines that the integrand basis contains 32 terms. In terms of (3.2), the basis can be presented as, On the cut, the integral becomes a meromorphic one-form, where N is a polynomial in α 3 , α 4 , β 3 and β 4 , and therefore also an elliptic function. We now perform the integral reduction, following the strategy as in the previous section. The task is to find exact meromorphic one-forms ω on this elliptic curves. If two integrals on the cut, differ by the contour integrals of such an ω, then where the second equality holds for all contours, i.e., two fundamental cycles and small contours around the poles, because ω is exact. Then the integral reduction between I[N 1 ] and I[N 2 ] is achieved at the level of double box diagram, Note the α 4 and β 4 generate all elliptic functions on this elliptic curve, as shown in the appendix, (A.5).
In practice, we find that to find such ω's, it is sufficient to consider the exterior derivatives of polynomials in α 3 , α 4 , β 3 and β 4 , So we need to find the one forms {dα 3 , dα 4 , dβ 3 , dβ 4 } and then use the chain rule to generate integral reduction relations. We can start by calculating dα 4 in terms of the holomorphic one-form, where we used the definition η = √ ∆ and (3.7) to rewrite η in function of β 4 . The purpose of this step is to get the a polynomial form of f .
We can now easily find dα 3 , the constant λ 1 is the product of α 3 α 4 on the cut. To generate the remaining 1-forms, we again use the form of elliptic curve. Recall that, The identity dK = 0 reads, Finally we can easily calculate dβ 3 , Then use the chain rule, we get all the on-shell IBPs. For example, from (3.16), we analytically obtain this relation, where . . . stands for integrals with fewer than 7 propagators. Consider all polynomials whose exterior derivative satisfy the renormalizability conditions, we obtain 23 integral relations. Furthermore, consider Levi-Civita insertions which integrate to zero, ǫ(l 2 , k 1 , k 2 , k 3 ) l 2 · k 1 , ǫ(l 2 , k 1 , k 2 , k 3 ) l 1 · k 4 , ǫ(l 1 , l 2 , k 1 , k 2 ) , ǫ(l 1 , l 2 , k 1 , k 3 ) . and explicitly the MIs can be chosen as, . (3.25) or in the conventional choice with X ≡ (l 1 + k 4 ) 2 /2 and Y ≡ (l 2 + k 1 ) 2 /2, 26) and for instance, the integral reduction in this basis becomes, The whole computation takes about 120 seconds with our Mathematica code. The relations are numerically verified by FIRE [3,4].

Reduction of the double-propagator integrals
One issue not discussed in [31] is the reduction of integral with internal mass and doubled propagators. For the double box diagram, the doubled-propagator integral on the cut also becomes meromorphic 1-forms, so we may carry out the maximal unitarity analysis as that in [31]. However, in this section, we show that, our new method is more efficient for reducing these integrals. Consider the diagram in Fig. 1 with the middle propagator doubled, (3.28) on the cut, by the degenerate residue computation with transformation law or Bezoutian matrix computation [48,49], we have, Unlike the one-forms in the previous subsection, here the one-form have the denominator ∆ 3/2 . It implies that we need to find exact 1-forms like d(F/∆ 1/2 ), where F is a polynomial in the loop-momenta components. Note that ∆(a 4 ) has four distinct roots, hence ∆(α 4 ) and ∆ ′ (α 4 ) have no common root. By Bézout's identity, ∆(a 4 ), ∆ ′ (a 4 ) = 1 (3.30) and the polynomial Diophantine equation has solutions. Such polynomials f 1 and f 2 can be explicitly found by Euclidean division or Gröbner basis method. The exterior derivative, determines that, after integrating out the exact form. Now the term ∆ 3/2 is removed and we can reduce this integral using the result from the previous subsection. In practice, we find a solution such that f 1 + 2f ′ 2 is a quadratic polynomial in α 4 , so at the level of the double box, The three coefficients c 0 , c 1 and c 2 are analytically found by our method and numerically verified by FIRE [3,4].

Elliptic Example: Sunset Diagram
The sunset diagram is a two-loop diagram which attracts a lot of research interests . The sunset diagram with 3 different masses is the simplest loop diagram which cannot be expresses in multiple polylogarithms. We use our method to study the integral reduction of the sunset diagram (Fig. 2) in two dimensional space-time. In this example, we assume that all internal propagators are massive. Let p be the external momentum, p 2 = m 2 . We can parametrize the loop momenta using a variant of Van-Neerven Vermaseren basis [73]. Define two null vectors e 1 and e 2 such that e 2 1 = 0, e 2 2 = 0 and e 1 · e 2 = p 2 . The Gram matrix of {e 1 , e 2 } is, In this basis we expand p as, and define the auxiliary vector ω, Hence p · ω = 0. The two loop momenta can be then generally parametrized as On-shell equations are D 1 = D 2 = D 3 = 0, where D i represent the inverse propagators, The on-shell solution can be formally expressed as, where again α 2 and β 2 satisfy the equation of an elliptic curve A(α 2 )β 2 2 +B(α 2 )β 2 +C(α 2 ) = 0. The discriminant is ∆ = B 2 − 4AC.
Define that X = l 1 · p and Y = l 2 · p, the four master integrals can be chosen as, For instance, the reduction reads, where . . . stands for integrals with fewer than 3 propagators. Note that generically, D-dimensional sunset diagrams with 3 distinct masses have 4 master integral. The four master integrals can be chosen as (4.15) or the scalar integral and three doubled-propagator integrals. There is a subtlety that if D = 2, then the 4 master integrals are related by Schouten identities [69]. These identities are valid for D < 3, and at D = 2 they further reduce the number of master integrals from 4 to 2.

Hyperelliptic Example: Nonplanar Crossed Box with Internal Masses
We now proceed in studying the integral reduction of the massive nonplanar double box (Fig. 3). Unlike the previous examples, this diagram's maximal unitarity cut provides a genus-3 hyperelliptic curve [46,47]. The structure of holomorphic/meromorphic forms on this curve is different from the elliptic case. However, our new approach for integral reduction works for this case as well.
To illustrate our method, we consider the two-loop crossed box diagrams with massless external legs and three internel masses scales {m 1 , m 2 , m 3 }. Our method also works for other crossed box configurations with all massive internal legs.

Maximal Unitarity and geometric properties
The denominators for the Fig. 3 are, The on-shell constrains are We use the same loop momenta parametrization (3.2). Again, we first solve for α 1 , α 2 , α 3 , β 1 , β 2 and β 4 in terms of α 4 and β 3 , The rest two variables satisfy a polynomial equation, whose solution can be formally represented as, Unlike the previous examples, ∆(α 4 ) here is a degree-8 polynomial in α 4 with 8 distinct roots. Hence the unitarity cut of this diagram provides a genus-3 hyperelliptic curve. (See the appendix for the classification of complex algebraic curves). Note that not all genus-3 algebraic curves are hyperelliptic, but this one is because of (5.4). Before the integral reduction, it is interesting to the see the geometric properties of this unitarity cut. Using (5.4) and the statements of appendix, we see that the function α 4 is a meromorphic function of degree 2 on the curve, i.e., it has two poles P 1 , P 2 and two zeros Q 1 , Q 2 . Explicitly, we can check that Q 1 and Q 2 are distinct, therefore, in the language of divisors (A.7), The divisor of α 3 is then The divisor for the function β 3 and β 4 is more complicated. From (5.4), we determined that β 3 on the cut becomes a meromorphic functions of 4 simple poles. The divisor of β 3 is: We find that two poles of α 4 become a pole and a zero of β 3 . Similarly, the two zeros of α 4 also become a pole and a zero of β 3 . The divisor of β 4 is: In summary, there are 8 poles on this hyperelliptic curve from numerators insertions, namely P 1 , P 2 , Q 1 , Q 2 , Z 1 , Z 2 , W 1 and W 1 .

Integral reduction
First, the integrand reduction via Gröbner basis [40,41] determines that, the integrand basis contains 38 terms in the numerator, Then, consider the maximal cut for the scalar integral of this diagram. The residue computation gives, Note that unlike the elliptic case, on a genus-3 curve there are three holomorphic 1-forms from (A.13), (which have no pole on the hyperelliptic curve), the scalar integral cut corresponds to the second one, while It is curious that for a crossed box integral with the numerator linear in l 1 , the maximal cut always gives a holomorphic 1-form. This hyperelliptic curve have 6 fundamental cycles and 8 poles as shown in the previous subsection. By global residue theorem, only 7 poles' residues are independent. Therefore we may perform maximal unitarity by computing integrals over 6 + 7 = 13 contours. Since here we only have one unitarity cut solution, the number of master integers must be less than or equal 13. However, in the following discussion, for the purpose of the integral reduction, we use our new approach to find exact meromorphic forms instead of calculating these integrals explicitly.
Following what we did for elliptic cases, we would like to generate the IBP relations by exact meromorphic 1-forms on the hyperelliptic curve. Again, we calculate the differential forms {dα 3 , dα 4 , dβ 3 , dβ 4 }.
where we have used the usual definition η ≡ √ ∆. In the second equality, we used the on-shell identity, to recover the form of the scalar integral cut (5.11). The step is not needed for elliptic cases. Then, where again we have used (5.3) to simplify our expression. The exterior derivatives for β i are more complicated, and, Given a polynomial function of {α i , β i }, we can use the chain rule to generate the on-shell IBPs.

Conclusions
In this paper, we present the relation between the on-shell IBPs and the meromorphic oneforms on algebraic curves, for a class of two-loop diagrams: D-dimensional L-loop diagram with DL − 1 propagators and one unitarity cut solution. In this case, the unitarity cut has a globally well-defined one-dimensional complex structure on it, and hence the analysis of IBPs relations can be translated into the analysis of complex curves.
By presenting several two-loop examples, planar and non-planar, we show that from the knowledge of algebraic curves, it is very easy to construct an IBP relation which reduces an arbitrary integral to master integrals. No explicit form of elliptic/hyperelliptic function is needed in our method, since only the differential relations of these functions are needed. Our method works for the reduction of integrals with or without doubled propagators.
There are several interesting future directions. In this paper, we mainly consider diagrams of DL−1 propagators without three-point massless vertice. If a (DL−1)-propagator diagram has three-point massless vertices, generically, the unitarity cut is not an irreducible curve but a reducible curve, i.e, union of several irreducible algebraic curves. We find that the IBPs obtained from our method, has a smooth massless limit and the limit forms a subset of IBPs for these diagrams. For example, the massless limit of our method, applied on the massless double box diagram, provides all but 2 IBPs. The missing 2 IBPs contain only low-rank numerators, so can be easily found by other algorithms. So in these cases, our method would greatly speed up the integral reduction process, even if IBPs are not all obtained. In the future, we expect that the algebraic geometry analysis on reducible curves will lead the complete set of on-shell IBPs of these diagrams.
Furthermore, we may consider using geometric properties of algebraic surfaces to study loop diagrams with an arbitrary number of propagators. It is well known that the algebraic geometry property of surfaces is more complicated than that for curves. However, we expect our approach will be generalized for the surfaces cases, because essentially our method does not depend on the detailed information of elliptic/hyperelliptic functions or integrals. Only the complex structure and differential relations are needed. So the surface cases would be studied following this direction, and to recover the ". . ." terms in our reduction like (3.27) and (5.21).
Finally, we will study the ǫ-dependent part of the integral reduction, based on our method. In this paper, we consider diagrams with integer-valued spacetime dimension. The ongoing research on two-loop maximal unitarity in dimensional regularization scheme [74], also based on algebraic geometry tools, will help us to understand IBPs with dimensional regularization from a geometric viewpoint.

A Rudiments of Algebraic Curves
In this appendix we give a brief introduction to the mathematical background of algebraic curves used in this paper. The extensive treatment can be found in ref. [33][34][35].
Definition 1 A Riemann surface is a connected one-dimensional complex manifold.
We are mostly interested in compact Riemann surface. Any compact Riemann surface is homeomorphic to a sphere with g ≥ 0 handles attached, and the number g is called the genus of the Riemann surface. Since the complex dimension is one, we also denote a compact Riemann surface as a complex algebraic curve. However, rigorously speaking, we may need to blow up possible singular points on an algebraic curve to make it a Riemann surface.
Definition 2 A holomorphic map between Riemann surfaces X and Y is a continuous map f : X → Y such that for each holomorphic coordinate φ U on U containing x on X and ψ W defined in a neighbourhood of f (x) on Y , the composition is holomorphic.

Definition 3
A meromorphic function f on a Riemann surface X is a holomorphic map to the Riemann sphere S = C ∪ {∞}.
One very useful theorem regarding to the topological properties of algebraic curves is Riemann-Hurwitz theorem. Here we consider the special cases of f : X → S. If in a neighborhood of P ∈ X, P is located at the origin and f has the expansion f (z)−f (0) ∼ z n , n > 1, then we say P is a ramified point of f and n is the ramification index of P . Removing images of ramification points under f , we get a Riemann sphere excluding a finite number of points, namelyŜ. For any point Q ∈Ŝ, define that d(Q) ≡ #f −1 (Q), the number of points in the inverse image. d is an integer-valued and continuous function, hence a constant. This constant d is called the degree of f . For a compact Riemann surface, the Euler character is related to the genus g, i.e., number of handles as, χ = 2 − 2g. (A. 3) The g = 0 compact Riemann surface is a Riemann sphere, while g = 1 compact Riemann surface is an elliptic curve (or torus topologically). g > 1 cases are more complicated, and we focus a particular class, hyperelliptic curves, which is defined as an algebraic curve, h is a degree-n polynomial in x with n distinct roots. C has the genus g, if d = 2g + 1 or d = 2g + 2, by (A.2). Note that g = 2 curve must be hyperelliptic, but not all g > 2 curves are hyperelliptic.
In the hyperelliptic case, x and y become meromorphic function with these properties, • x is a meromorphic function of degree 2 on C, • y is a meromorphic function of degree n on C, • x : C → S has 2g + 2 ramified points. If n is even, these points are (x, y) = (a i , 0) where a i 's are the roots of h(x). If n is odd, these points are (x, y) = (a i , 0) and the point at infinity.
• Every meromorphic function f on C can be uniquely written as f = r(x) + ys(x) (A.5) where r(x) and s(x) are rational functions of x.
The last property (A.5) is important for studying the exact meromorphic 1-forms, which play the central role of our integral reduction algorithm.

A.1 Riemann-Roch theorem
We now want to state one of the fundamental theorems of compact Riemann surface X. First, we present several definitions, We can naturally associate a divisor to a meromorphic function f in the following way, If ω is a meromorphic differential on a Riemann surface X then the number of zeros of ω minus the number of poles, counted with multiplicity is 2g − 2.
For the hyperelliptic curve (A.4), we can find an explicit basis of the holomorphic one-forms, form a basis of holomorphic differential forms.
We use this basis frequently in our paper for the integral reduction.