2. On a Conjecture of Schur, Michigan Math. J. Volume 17, Issue 1 (1970), 41–55 (pdf also on-line at the Michigan Math Journal). My first paper, though not first in print. It gives the classification of exceptional polynomials – those that map one-one on infinitely many residue fields – of a number field. They are up to (very precise) linear change over the algebraic closure compositions of cyclic (like xⁿ) and Chebychev polynomials. Schur's 1921 Conjecture generated much literature: at its solution Charles Wells sent me a bibliography of over 550 papers, most showing certain families of polynomials – given by the form of their coefficients – contained none with the exceptionality property. An essential step was recognizing and using a reduction to polynomials with primitive monodromy group.

Includes the first serious use of R(iemann)'s E(xistence) T(heorem) on a problem of this type, a start of the monodromy method. Chebychev covering groups are dihedral and easy to characterize. So, RET was quick, but not essential here. Yet, Schur's Conjecture was special, and much easier, within Davenport's problem, and RET has proved essential for that. Still, by considering its analog for rational functions, the monodromy method connected to Serre's O(pen)I(mage)T(heorem) (UMStoryExc-OIT.html and GCMTAMS78.pdf) and, so, to modular curves. Further, using RET opened the territory to many other problems (especially see the complete story of Davenport's problem in UMStory.pdf). SchurConj70.pdf

3. The field of definition of function fields and a problem in the reducibility of polynomials in two variables, Illinois Journal of Math. 17, (1973), 128–146. The pdf file is a scan. The paper's center is the solution of Davenport's Problem. Variables separated Equations, contrasts the contributions of the simple group classification and of the Branch Cycle Lemma (for figuring the defining field of an algebraic relation) and the other problems these techniques influenced through the monodromy method.

Davenport's problem was essentially to classify polynomials over Q by their ranges on almost all residue class fields. The most general results, restricted to polynomials not composable (indecomposable) from lower degree polynomials, gave two very different conclusions:

1. Over Q two such polynomials with the same range are linearly equivalent: obtainable, one from the other, by a linear change of variables.
2. For certain number fields polynomials that aren't linearly equivalent could have the same ranges for all residue class fields. Yet, there are only finitely many exceptional degrees, and we understand them.

• § 7.2 of thomp-genus0.pdf has its history and relation to this paper.
• § 2.4 contains Peter Müller's list of primitive polynomial monodromy groups – showing the large contribution of the monodromy from Davenport pairs – a tour-de-force in its own right.

4. On the Diophantine equation f(x) -y = 0: Acta Arith. XIX (1971), 79–87. On the interface between Davenport's (comparing value sets of polynomials on residue classes) and Schinzel's (on factorizations of f(x)-g(y)) problems, and related to the discovery of their near equivalence when f is indecomposable. Though without the depth or breadth of dav-red.pdf as related in UMStory.pdf (describing the many consequences of it in the work of others), it has one big virtue: Without any restriction on indecomposability it treats the case of odd prime-power degrees for polynomials over Q. This was a rare case of circumventing assuming a cover has a primitive monodromy group. ActArith71f-x.pdf

5. On a Theorem of MacCluer, XXV Acta Arith.~(1974), 121–126. The Cebotarev density theorem applied to a cover over a finite field often translates diophantine statements into statements about the monodromy group of the cover. Usually it is somewhat clumsy; you can't conclude the monodromy statement implies the original diophantine condition, except off the ramification locus. Sometimes, however, you can use it to give a valuable Monodromy Converse.

Consider a cover of the projective line over a finite field F given by a polynomial f. Take the fiber product of the cover given by f with itself, remove the diagonal component and normalize the result in its function field. Denote this f^~₂. Refer to f as exceptional if f^~₂ has no absolutely irreducible components. This definition applies to any cover of any space over any field.

MacCluer's thesis in 1967 answered the main question of a Davenport-Lewis 1966 paper by showing that if f is a tame polynomial, then f exceptional implies f is (precisely) one-one as a map on F, and therefore on infinitely many extensions of F. This paper interprets a generalization of this as a monodromy converse – now called monodromy precision: You can determine the diophantine property just from a statement on the monodromy of the cover. This paper generalizes MacCluer's theorem to drop the tame condition and that the map is of dimension 1 covers. Eventually, a far more general condition called p(ossibly)r(educible)-exceptional was shown to satisfy monodromy precision: Cor. 3.6 of exceptTower0910-3331v1.pdf. As a special case, this applied to generalized Davenport pairs. MacTheom74.pdf

6. Arithmetical properties of function fields (II): The generalized Schur problem: Acta Arith. XXV (1974), 225–258. Using the analog of Schur's conjecture over a finite field, this paper introduced a new approach to understanding wildly ramified covers of the projective line. The outstanding point about the monodromy version of the Schur property is the significance of distinguishing between the arithmetic and geometric monodromy of a cover. The main topic of § 1 is the introduction of ramification data for wild ramification, extending the notion of higher ramification groups for not necessarily Galois covers. While not so general (and with limited use of families) as the treatment in Configuration Spaces for Wildly Ramified covers, it has many helpful examples. Again, with many examples, the § 2 is a forerunner of Galois groups and Complex Multiplication. genSchur74.pdf

7. with G. Sacerdote, Solving diophantine problems over all residue class fields of a number field ..., Annals Math. 104 (1976), 203–233. Picked up (with Ken Ribet's help) from JStor http://links.jstor.org/sici?sici=0003-486X%28197609%292%3A104%3A2%3C203%3ASDPOAR%3E2.0.CO%3B2-P. Introduces the Galois Stratification procedure in its Original, geometric form. Corresponds roughly to the non-geometric approach of Chap. 25 of the Fried Jarden book (1986 edition; Chap. 30 in 2005 edition). Chap. 26 (resp. Chap. 31) includes the start of zeta function applications developed in an untexed preprint, "L-series on a Galois Stratification," I spoke on in Spring, 1979 Lectures at Yale. How the paper works.

Elementary statements encode many of the famous (old included) problems in mathematics, like Chevalley's Theorem: Over any finite field, F_q, for each degree d, each form of degree d in projective d-space, has an F_q point. It was the Ax-Kochen work on the p-adic Artin version: Over any p-adic field Q_p, each form of degree d² in projective d-space, has a Q_p point, and the complication of its almost true nature that motivated the general problem of this paper.

That is: Is there a useful procedure for deciding statements that generalize such classical problem, at least for a fixed d? The secret is, working in blocks of variables, each quantified by "there exists" or "for each," to generalize what is an elementary statement, so that without messing with the blocks, they can be eliminated one at a time. The generalization is called a Galois Stratification,. The special tool is the generalization of the Chebotarev Density Theorem. The quantifications are of variables in the base of a finite cover whose Frobenius is contained in a specific union of conjugacy classes in the Galois group of the cover. Galois stratifications were suitable coefficients for attaching a zeta function to essentially any diophantine question over a finite or p-adic field. We used The first step in the solution of Davenport's problem to illustrate just how effective is this procedure. annals76.html %-%-% annals76.pdf

8. with D. Haran and M. Jarden, Galois Stratification over Frobenius Fields, Advances in Mathematics 51 , 1–35 (1984). The Galois stratification procedure was introduced in Solving diophantine problems over all residue class fields of a number field and all finite fields, Ann. of Math. 104 (1976) 203–233. It was the first elimination of quantifiers procedure that applied to problems beyond Tarsky's method for eliminating quantifiers over the field of real numbers.

It gave an explicit quantifier elimination over such systems fields as all residue class fields of a number field. Further, it produced zeta functions attached to general diophantine statements. The trick was to use Galois stratifications as coefficients of formal zeta functions to which a Frobenius element could be applied. One result was that such zeta function were close to (quasi-) rational functions, for which one could explicitly compute their parameters, including the polynomials of their numerators and denominators. The method easily extended to consider various problems over all completions of a number field.

Eventually Denef and Loeser modified that to replace the Galois stratifications by Chow motive coefficients. For some situations this combined with Hironaka's resolution of singularities to produce zeta functions canonically attached to the diophantine statement.

This paper extends the application of Galois stratification to systems of fields with a resemblance to systems of finite fields. These include, for e a finite positive integer, the PAC e-free subfields of the absolute numbers. The P(seudo)A(lgebraically)C(losed) property holds if all absolutely irreducible varieties over such fields have rational points. Almost all e-Free subfields have that property. For such fields there is a natural replacement for the Chebotarev density theorem that thereby extends the Galois stratification procedure. GaloisStratoverFrobFields1984.pdf

9. L-series on a Galois Stratification: I decided to put this on-line in response to a request from someone who had read the Science China paper (below). This was written in the 70s, during the time I had tried to explain to Denef the potential for more refined coefficients attached to zeta functions using a refinement of Galois stratifications. Denef didn't understand Chebotarev (logicians at the time did not), but Loeser – when we were both giving lectures at Ihara's 65th birthday celebration in Tokyo – explained how he had come to understand what I was doing. I discuss this more in the Science China paper.

Denef-Loeser used Galois stratification to illuminate quantifiers almost straight from my paper. Then, however, instead of a flattening stratification, as I had done, they applied Hironaka's resolution of singularities, to get Chow motives, an ineffective process. So, they have no control on exceptional primes. By doing this they were able to replace my zeta functions with Galois stratification coefficients (to which you specialize the Frobenius) with Chow motive (an expression in linear combinations of l-adic cohomology of projective nonsingular varieties) coefficients. So, they didn't produce Galois stratifications over arbitrary primes, while my procedure did.

On the other hand, even to me, this paper seems old-fashioned. The category of Galois stratifications is not efficiently defined. Denef-Loeser's use of Chow motives produced zeta functions with properties that gave diophantine expressions properties like those of nonsingular algebraic varieties. This p-adic paper was aimed at using L-Series for a refined definition of exceptional primes that would work on all diophantine expressions. The Chow motive approach came much later.

Nicaises' paper, referenced in Science China paper was based on Vovoidsky's theory of motives. I'm not up on that, and Nicaise himself struggled with it (I caught an error in his original version of his paper). That doesn't, however, answer the question above.

Also, there is the work of Tomasic based partly on Hrushovski and Chatzidakis, and partly on work that was entirely new to me on correspondences. These took up the idea of diophantine expressions over fields with a distinguished automorphism. It was a model theory approach extending Galois stratification over, say, all finite field extensions of a given finite field to such expressions over the algebraic closure, with a predicate for the Frobenius.

If you aren't a model theorist, well, I'm not either, but it was a generalization. I discuss that in my referee report of Tomasic. LSeriesGalSt86.pdf

10. with R. Lidl, On Dickson Polynomials and Rédei Functions, Contributions to General Algebra 5, Proceedings of the Salzburg Conference, Mai 29 - June 1,1986 Verlag Holder-Plchler-Tempsky, Wien 1987 - Verlag B. G. Teubner, Stutgart, 139–149. Recall: An exceptional cover f: P¹ → P¹ over a number field K is one for which f maps one-one on the points for infinitely many residue class fields (with ∞ adjoined) of K. Denote the set of primes for this set by E'_f. There is also a notion of exceptional over a given finite field. The file n-dimChebychev.pdf explains this, definition of the exceptional set E_f of f (that E'_f minus E_f is a finite set), and the tie to the modern theory of exceptional covers of arbitrary dimension.

Especially it includes basic properties of Chebychev covers of Pⁿ⁺¹ having degree k generalizing the usual 1-dimensional Chebychev polynomials. It is essentially this paper's construction as applied to higher versions of Dickson polynomials. Though it was written 20 years before , The place of exceptional covers among all diophantine relations, J. Finite Fields 11 (2005) 367–433, arXiv:0910.3331v1, recent work of others – referenced in n-dimChebychev.pdf – suggests it is time to resurrect the construction and test its generality. The most attractive problems in this area ask for descriptions of the variants of E_f. They represent an explicit challenge to the Langlands Conjecture for producing L-series with Euler products that affect problems with a (very) large literature; one that can feed off the exceptional rational functions that come from Serre's Open Image Theorem. Dickson-Redei86.pdf

11. with R. Guralnick and J. Saxl, Schur Covers and Carlitz's Conjecture, Israel J.; Thompson Volume 82 (1993), 157–225: We use the classification of finite simple groups and covering theory in positive characteristic to do much more than solve  Carlitz's conjecture (1966). An exceptional polynomial f over a finite field F_q is a polynomial that is a permutation polynomial on infinitely many finite extensions of F_q. Carlitz's conjecture says f must be have odd degree (if q is odd). It is immediate that you can reduce to considering the case that f is indecomposable over F_q.

The more precise results are as follows. Recall an affine group – in what appears below – is a semidirect product of a vector space V over F_p and a subgroup of the general linear group GL(V) acting on V.

• Excluding characteristic 2 or 3, arithmetic monodromy groups of exceptional polynomials must be affine groups with the degree the cardinality of V.  
• We completely classify exceptional polynomials of degree equal to the characteristic, an unsolved problem from Dickson's thesis (1896). Further, by generalizing Dickson's problem we describe all known exceptional polynomials.
• Further, by generalizing Dickson's problem we describe all known exceptional polynomials. Especially this includes Cohen's semi-linear polynomials. They all fit a simple criterion in our use of monodromy groups. 
• In the case of characteristic 2 and 3 we show precisely what those arithmetic monodromy groups must be, but we didn't try to find out if there were polynomials that realized those particular groups.

12. with S. Cohen, The Carlitz-Lenstra-Wan conjecture on Expectional Polynomials: An Elementary Version: Finite Fields and their applications, Carlitz volume 1 (1995), 372–375. If you want to be able to algebraically scramble data embedded as an element in an arbitrarily large finite field while fixing the scrambling function, then you must use an exceptional rational function as scrambler. Finding exceptional polynomials (they fix the point at ∞) is a piece of that, and [FGS] comes close to it. The much weaker Lentra-Wan Statement – proved here – says exceptional polynomials F_q have degrees prime to q - 1. The html file explains just how weak is that statement. carlitz-quick.html %-%-% carlitz-quick.pdf

13. Global construction of general exceptional covers, with motivation for applications to coding, G.L. Mullen an P.J. Shiue, Finite Fields: Theory, applications and algorithms, Cont. Math. 168 (1994), 69–100. globConstExcCov.html %-%-% globConstExcCov.pdf

14. with D. Haran and M. Jarden, Effective counting of the points of Definable Sets over Finite Fields, Israel J. of Math 85 (1994), 103--133. Galois stratification provides here (as initiated in Solving diophantine problems over all residue class fields of a number field … Annals Math. 104 (1976) 203–233 an effective procedure for eliminating quantifiers in 1st order equations over various collections of fields. For example, all residue class fields of a number field, or all finite fields F_q, q running over all powers of primes.

The main theorem starts with any formula φ in variables x₁, …,x_m; y₁, …,y_n, in the language of rings. It explicitly produces these:

1. an integer k; integers r₁, …,r_k between 0 and n; positive rational numbers μ₁, …,μ_k, computed by counting elements in conjugacy classes of Galois ring covers; and
2. a finite collection of formulas φ₁(x), …,φ_k(x) so that for each a in (F_q)^m the following holds.
• If φ( a;y) holds in F_q, then there is a unique i for which φ_i( a) holds in F_q;
• with N_q,i the number of a in (F_q)^m for which φ_i(a) holds,
• either N_q,i is 0 or the ratio of | N_q,i - μ_iq^r_i| and q^r_i-1/2 is bounded in q.

15. Applications of Curves over finite fields, in Curves over Finite Fields Cont. Math., proceedings of AMS-NSF Summer Conf. 1997, Editor M. Fried, Seattle 245 (1999), ix–xxxiii: This is an exposition on the themes in the papers presented at the conference. Starting from the role of Deligne's proof of the Weil Conjectures and the classification of finite simple groups, the sections divide the conference papers into practical tools. As was the last conference attended by Bernie Dwork, the final section includes comments on his work that complement an article of Katz and Tate.

1. Beyond Weil bounds; curves with many rational points: The moduli space approach; The Drinfeld module approach when q is not a square; More on Explicit use of Drinfeld modules; One curve with many points and fiber products; Approach from classical curves.
2. Monodromy groups of characteristic p covers: What to expect of monodromy groups from genus 0 covers; Abhyankar's approach; Reflection on classical invariant theory; Reduction mod p and field of moduli of covers; Refined abelian covers; Good reduction of covers; Explicit computation of monodromy grops over finite fields.
3. Zeta Functions and Trace Formulas: Unit root L-functions; Zeta functions of complete intersections; Properties of a modular curve quotient; Appearance of rank 1 representations in L-functions; Eigenvalues of a Laplacian; Average value of Zeta-functions and elliptic surfaces
4. A Dedication to the Work of Bernie Dwork: Michael Rosen: Dwork's relation to his students; Pierre Dèbes: Dwork's role in G-functions; Alan Adolphson: Dwork's final Conjecture.

16. with W. Aitken and L. Holt, Davenport Pairs over finite fields, PJM 216, No. 1 (2004) 1–38. davpairs07-22-04PJ.html %-%-% davpairs07-22-04PJ.pdf

17. with Ariane Mézard, Configuration Spaces for Wildly Ramified covers, in Proceedings of Symposia in Pure Mathematics 70 (2002) editors M. Fried and Y. Ihara, 1999 Symposium on Arithmetic Fundamental Groups and Noncommutative Algebra, August 16-27, 1999 MSRI, 353–376: A natural configuration space – the space of unordered branch points of a cover – for tame covers of curves allows the construction of Hurwitz spaces, and an effective theory for families of tame covers of curves. We define invariants generalizing Nielsen classes for Hurwitz families to produce a configuration space for classifying families of wildly ramified covers (in positive characteristic).

We don't assume the covers are Galois. The center of this construction is what we call local ramification data, a generalization of the upper numbering of higher ramification groups. The potential of the method appears with problems posed by the construction of the Galois closure of a family of covers with r distinct branch points. This extends to wildly ramified covers the first half of Grothendieck's Theorem describing tamely ramified covers. The html file exposits on these topics:

1. Motivation from Hurwitz spaces
2. Local Ramification Data
3. Global Configuration Spaces
4. The Major Unsolved Problem

18. The place of exceptional covers among all diophantine relations, J. Finite Fields 11 (2005) 367–433, arXiv:0910.3331v1 [math.NT]; short list of corrections at exceptTower0910-3331v1-cor.html. A cover of normal varieties is exceptional over a finite field if the map on points over infinitely many extensions of the field is one-one. A cover over a number field is exceptional if it is exceptional over infinitely many residue class fields. The first result: The category of exceptional covers of a normal variety Z over a finite field, F_q, has fiber products, and therefore a natural Galois group (with permutation representation) limit. This has many applications to considering Poincare series attached to diophantine questions. The paper follows four lines:

• The historical role of the Galois Theoretic property of exceptionality, first considered by Davenport and Lewis.
• How the tower structure on the category of exceptional covers of a pair (Z,F_q) allows forming subtowers that separate known results from unknown territory.
• The use of Serre's OIT, especially the GL₂ case, to consider cryptology periods and functional composition aspects of exceptionality.
• The expansion of the fiber product definition of exceptionality to p(ossibly)r(educible)-exceptionality, thereby including Davenport pairs and others satisfying monodromy precision in identifying solutions.

19. The place of exceptional covers among all diophantine relations, J. Finite Fields 11 (2005) 367–433. This is the original journal article, with pencil corrections that have been incorporated in arXiv:0910.3331v1 [math.NT]. exceptTowYFFTA_519.pdf

20. Variables Separated Equations and Finite Simple Groups: (abridged) What I learned from graduate school at University of Michigan, 1964–1967 The story of the monodromy method, as told by recounting the solution of Davenport's Problem. A longer version attached to it discusses its influence on the following projects:

1. Translation between the Davenport-Lewis conjecture on exceptional covers and Serre's Open Image Theorem.
2. Applying the simple group classification to the genus 0 problem (conversations with Feit, McLaughlin and Thompson).
3. the Galois stratification forerunner of Chow motives (from my first Annals paper solving the Ax-Kochen Problem).

21. Variables separated equations: Strikingly different roles for the Branch Cycle Lemma and the Finite Simple Group Classification: arXiv:1012.5297v5 [math.NT] (DOI 10.1007/s11425-011-4324-4). Science China Mathematics, vol. 55, January 2012, 1–72 (Journal article is here UMStory022011-70-120001.pdf.) Davenport's problem is to figure out the nature of two polynomials over a number field having the same ranges on almost all residue class fields of the number field. Solving this problem initiated the monodromy method. That included two new tools: the B(ranch)C(ycle)L(emma) and the Hurwitz monodromy group. By walking through Davenport's problem with hindsight, variables separated equations let us simplify lessons on using these tools. Many papers have quoted small pieces of the original paper dav-red.pdf. Yet, few used the full power of the method, so we revisit the works of many authors. We also attend to these general questions:

1. What allows us to produce branch cycles from covers, and what was their effect on the Genus 0 Problem (of Guralnick/Thompson)?
2. What is in the kernel of the Chow motive map, and how much is it captured by using (algebraic) covers?
3. What groups arise in 'nature' (a 'la a paper by R. Solomon)?

22. Schur's Conjecture and Exceptional Covers, § 9.7 in Handbook of Finite Fields, (2013), 290–302, editors G. Mullen and D.~Panario. HFFTContents.pdf is a complete table of contents. Serre's O(pen) I(mage) T(heorem) gets much credit as a sophisticated piece of mathematics, but its practical uses are little known. This paper collects observations over the years that rational functions in one variable are a center of problems in the work of many mathematicians, through compositions of rational functions, and more generally, sequences of covers. We specifically clarify the connection between the OIT and the complete description of the following covers, revealed through the monodromy method:

1. rational function exceptional covers over number fields; and
2. rational functions over a number field or finite field that are indecomposable, but decompose over an algebraic closure.

23. Diophantine statements over Residue fields: Galois stratification and uniformity: To appear in the proceedings of the Fq14 conference in Summer 2019 in Vancouver. Using Felgner's problem this paper revisits a key issue in using the Galois Stratification Procedure that first appeared in 1976 in Solving diophantine problems over all residue class fields of a number field ..., Annals Math. 104 (1976), 203–233. We update considerable territory since then, especially emphasizing arithmetic homotopy aspects that make the production of Poincaré series attached to general diophantine statements canonical. Specifically we consider these two topics:

1. Testing how a quantified diophantine statement regarded over field Z/p coheres as a function of p.
2. Considering variables taking values in the algebraic closure of Z/p but fixed by respective powers of the Frobenius: we call these Frobenius vectors.

24. Enhanced Version: Diophantine statements over Residue fields: Galois stratification and uniformity: DOI: 10.13140/RG.2.2.25470.54081 Appeared in Finite Fields and their Applications, Proceedings of the 14th International Conference on Finite Fields and their Applications, Vancouver, June 3-7, 2019 Series: De Gruyter Proceedings in Mathematics Edited by: James A. Davis De Gruyter | 2020 DOI: https://doi.org/10.1515/9783110621730

The volume TOC: Solving diophantine problems over all residue class fields of a number field ..., Annals Math. 104 (1976), 203–233. The emphasis here is on using arithmetic homotopy to make the production of Poincaré series attached to general diophantine statements canonical.

According to work in progress of Michael Benedikt and E. Hrushovski, Galois stratification – over one finite field – is as efficient as is possible: on a statement of length n, it requires time bounded by a stack of exponentials, of length linear in n. This, however, doesn't take advantage of problems prepped for using homotopy aspects, Chow Motives, efficiently as in the main example which comes from my paper on the generalization of exceptional covers.

That example – a special case of considerable generality – exposits on technical details of Chap. 30 of Fried-Jarden, Springer-Verlag 2004, simplifying aspects of the original procedure. It combines this with the later theory of Frobenius fields to produce objects over Q whose reductions mod primes give the stratification procedure at the prime. The paper separates two different uses of the Chebotarev non-regular analog.

• Field crossing to interpret Poincaré series coefficients directly from traces on Chow motives (providing valuable statements on variation with p); versus
• Chebotarev using Lang-Weil to approximate the number of points on an appropriate variety for the Galois stratification procedure.

25. Taming Genus 0 (or 1) components on variables-separated equations: Preprint as of 08/06/22.

To figure the number theory properties of a curve of form C_f,g = {(x,y)| f(x) - g(y)= 0} you must address the genus 0 and 1 components of its projective normalization C^˜_f,g. For f and g polynomials with f indecomposable, [Fr73a] distinguished C^˜_f,g with u=1 versus u > 1 components (Schinzel's problem). For u = 1, [Prop. 1, Fr73b] gave a direct genus formula. To complete u > 1 required an adhoc genus computation.

[Pak22] dropped the indecomposable and polynomial restrictions but added C^˜_f,g is irreducible (u = 1). He showed – for fixed f – unless the Galois closure of the cover for f has genus 0 or 1, the genus grows linearly in deg(g). Method I and Method II extend [Prop. 1, Fr73b}] using Nielsen classes to generalize Pakovich's formulation for u > 1.

• Method I plays on the covers f and g to the z-line, P¹_z, from which we compute the fiber product.
• Method II uses the projection to the y-line, P¹_y, based on explicitly computing branch cycles for this cover.