# AMS Special Session: Algorithmic Algebraic and Analytic Geometry

## Atlanta, January 2005

Organized by Saugata Basu (Georgia Tech); Victoria Powers (Emory); Mika Seppälä (Florida State University); Tanush Shaska (University of Idaho); and Emil Volcheck (National Security Agency).

This special session will be held Friday and Saturday, 7-8 January 2005, at the Atlanta Joint Mathematics Meetings. This is the fourth special session on this area to be held at the AMS/MAA Joint Mathematics Meetings. The first special session was held in 1999 at the San Antonio Joint Meetings, the second special session was held in 2001 at the New Orleans Joint Meetings, and the third special session was held in 2003 at the Baltimore Joint Meetings.

### Scope

This session is devoted to algorithms and computational techniques for algebraic geometry, including algebraic curves, Riemann surfaces, algebraic surfaces, and low-dimensional varieties. We are interested in reports on algorithms to solve problems or on a significant use of computational algebraic or analytic techniques to obtain results. Algorithmic, algebraic, arithmetic, and analytic aspects are appropriate topics.

Read the list of speakers and their abstracts for the special sessions in 1999 and 2001 and 2003 on computational algebraic and analytic geometry to see what previous speakers in this series of special sessions have presented.

The final deadline for abstract submission was 5 October 2004.

## Speakers

The 22 speakers as of 3 January 2005 are

## Abstracts

### Large Sections of Nonnegative Polynomials and Sums of Squares

I have previously shown that for the case of a fixed degree there are significantly more nonnegative polynomials than sums of squares. However, I will explain that "well behaved" nonnegative polynomials have restrictions to subspaces of large dimension that are sums of squares. A conjecture of V. Milman says that every nonnegative polynomial should admit such a section.

### Remarks on the uniformizing function in genus 2

Peter Buser, Robert Silhol

Let $G$ be a Fuchsian group acting on the unit disk $\mathbb{D}$. Then $\mathbb{D}/G$ is an algebraic curve in a natural way. We describe, in genus 2, a practical way to compute the uniformizing function from $\mathbb{D}$ to this curve. The idea is based on a construction that modifies $G$ into a new group $G'$ having the same fundamental domain but being such that $\mathbb{D}/G'$ has genus 1, and then apply the classical uniformization of genus 1 curves.

### Extension of the Fourier-Budan theorem to one-variable signomials

Let $f(x)=a_0x^{r_0}+a_1x^{r_1}+\cdots+a_kx^{r_k}$, where each $a_i\in\mathbb R$, each $r_i\in\mathbb N:=\{0,1,\ldots\}$, and $r_00$. We also show that such an extension is impossible for arbitrary real analytic $f$.

### Linearly presented ideals and applications to embeddings of projective spaces

Any embedding of one projective space in another is given by an ideal of homogeneous polynomials, primary to the maximal ideal, generated a single degree, and with an additional property: some power of the ideal should be equal to a power of the maximal ideal. This property has surprising interactions with Groebner bases and homological properties of ideals. I will describe recent work with Bernd Ulrich and Craig Huneke on this topic. Sample theorem: any linearly presented primary ideal defines an embedding.

### Non-normal pairs of NEC groups

Milagros Izquierdo, Jose Luis Estevez

By a NEC (non-euclidean crystallographic) group we shall mean a discrete, cocompact subgroup $\Gamma$ of the group $\mathrm{Aut}(\mathcal{H})$ of all the automorphisms of the non-Euclidean plane $\mathcal{H}$. Given an NEC group $\Gamma$, we denote by $\mathbf{T}(\Gamma )$ the Teichmueller space of} $\Gamma$, it is homeomorphic to a real ball of dimension $d(\Gamma)$. We denote be $Max(\Gamma)$ the set of points in $\mathbf{T}(\Gamma )$ which represent maximal groups. $Max(\Gamma)$ is empty if there exists an NECgroup $\Gamma'$ such that $\Gamma \le \Gamma'$ and $d(\Gamma) = d(\Gamma')$. It is very interesting to study those NEC groups with empty $Max(\Gamma)$ because this fact helps us to determine whether a finite group $G$ can be the full group of automorphisms of a Klein surface $\mathcal{H}/\Gamma$ or not. Bujalance established these pairs $(\Gamma, \Gamma')$ when $\Gamma$ is normal in $\Gamma'$. In this paper we compute the pairs of such groups in the case when $\Gamma$ is non-normal in $\Gamma'$. The corresponding problem for Fuchsian groups was solved by Singerman.

### Effective methods for resolving singularities in the plane with applications to analysis

We describe an effective method for locally resolving the zero set of a real-analytic function $f(x,y)$. The method is geometric and involves doing a finite sequence of transformations of the form $(x,y) \rightarrow (x, y - g(x^{1 \over N}))$ for appropriate real-analytic functions g, where $N$ is an integer. After these transformations, a branch of the zero set of $f(x,y)$ will be (locally) given by $\{(x,y): x > 0, y = 0\}$ or $\{(x,y): x < 0, y = 0\}$. This method has applications to oscillatory integral operators, as well as to the determination of the largest $\epsilon$ for which $\int |f|^{-\epsilon}$ is finite near a given zero of a function $f(x,y)$.

### Fields of moduli of hyperelliptic curves

The field of moduli of a curve is, roughly speaking, the infimum of its fields of definition. Let $X$ be a hyperelliptic curve defined over a field $K$ of characteristic not equal to 2. Let $\mbox{Aut}(X)$ be the group of automorphisms of $X$ defined over an algebraic closure of $K$ and let $\iota$ be the hyperelliptic involution of $X$. We will present an overview of a proof that $X$ can be defined over its field of moduli if $\mbox{Aut}(X)/\langle\iota\rangle$ is not cyclic. We will also give examples of hyperelliptic curves not definable over their field of moduli when $\mbox{Aut}(X)/\langle\iota\rangle$ is cyclic.

### Modular representations on some Riemann-Roch spaces of modular curves X(N)

David Joyner, Amy Ksir

We consider the PSL(2,N)-module structure of the Riemann-Roch space L(D), where D is an invariant non-special divisor on the modular curve X(N), with n$>$5 prime. The first section reviews known results and gives, as an example, the cases N=7, 11. In the next section, ground fields of characteristic p$>$0 are considered. GAP and MAGMA were used extensively. Applications to AG codes associated to this curve are also considered. This paper ends with some tables, created using GAP, which yield computations for larger values of N.

### Birational Equivalence, Linear Systems, and Desingularization

John J Iskra, Yasuyuki Kachi, S B Mulay

In the talk I introduce an object Spv $X$ which represents the birational equivalence class of an algebraic variety $X$ and which admits a morphism to $X$. I define Spv ($X$) as a certain functor which mimics Hom (Spec (*), $X$) : (Ring) $\longrightarrow$ (Set). I also define its completion Spv ${}^{\wedge} (X)$, using linear systems, and show that it is the categorical limit of proper models birational to $X$. In the course it arises a group functor $SG_n$ which is a uniform analog of $GL_n$ and which reflects a composition algorithm of blow-ups. $SG_n (k)$ naturally acts on a certain classifying space of uniformizing parameters $\mathcal{S}_n (k)$. I show that the transitivity of such action is a uniform analog of Cutkosky's factorization theorem. Using $SG_n$, I also formulate a statement on constructibility of power series and show that it recovers the desingularization of an algebraic variety locally along a valuation.

Abstract as PDF

### Stiefel-Whitney classes of real toric varieties

The $k$th Stiefel-Whitney homology class of an $n$-dimensional real toric variety $X$ is represented by the mod 2 cycle which is the sum of the closures of the $k$-dimensional orbits of the action of $(\mathbb{R}^*)^n$ on $X$. To prove this result a theorem of Banchoff and the author is applied to the composition of the moment map $X \to \mathbb{R}^n$ with projection to a generic $(k+1)$-plane.

### On the extremal disks embedded in compact Riemann surfaces of genus three

Let $S$ be a compact Riemann surface of genus $g \ge 2$ equipped with the hyperbolic metric. Then $S$ is said to be extremal if a disk of radius $R_g$ is isometrically embedded in $S$, where $R_g$ is the maximal length determined by $g$. The disk embedded in $S$ is called an extremal disk. Our concern is the number and the position of the extremal disks that can be embedded in $S$. It was studied when $g\ge 4$ ([1]) and $g=2$ ([2, 3]). In this talk we shall consider extremal surfaces of $g=3$, and show that they have at most two extremal disks and that 16 of them (up to conformal equivalence) can admit exactly two.

References

E. Girondo and G. Gonz\'alez-Diez, On extremal discs inside compact hyperbolic surfaces, C. R. Acad. Sci. Paris, S\'er. I, Math. \textbf{329} (1999), no.1, 57--60.

E. Girondo and G. Gonz\'alez-Diez, Genus two extremal surfaces: extremal discs, isometries and Weierstrass points, Israel J. Math., \textbf{132} (2002), 221--238.

G. Nakamura, The number of the extremal disks embedded in compact Riemann surfaces of genus two, Sci. Math. Japon., \textbf{56} (2002), no. 3, 481--492.

### Hilbert's theorem on decomposability of nonnegative ternary forms as sums of squares of rational functions

A partial answer to the 17th Hilbert problem was obtained by Hilbert himself, in a famous paper [H1], and in much less well-understood paper [H2].

In [H2] it is proved that every nonnegative ternary form is decomposable as a sum (at most 4, as was observed later) of squares of rational functions. It was used recently in [dKP] to develop an algorithm, based on semidefinite programming, to calculate such decompositions on computer, prompting a need to understand [H2] better.

We discuss an approach leading to a modern proof for [H2], at least in some important cases. It is related to the recent work [PRSS] that contains a modern proof of a theorem from [H1] on decomposability of nonneg. ternary quartics as a sum of 3 squares.

References

[H1] D. Hilbert. \"Uber die Darstellung definiter Formen als Summen von Formenquadraten, Math. Annalen, 32(1888), 342-350.

[H2] D. Hilbert. \"Uber ternäre definite Formen, Acta Math. 17(1893), 169-198

[dKP] E. de Klerk and D.V. Pasechnik. Products of positive forms, linear matrix inequalities, and Hilbert 17-th problem for ternary forms, European J. OR, 157(2004) 39-45

[PRSS] V. Powers, B. Reznick, C. Scheiderer, and F. Sottile. Comptes Rendus, to appear. math.AG/0405475.

### Hilbert's wonderful eight-point construction

In 1888, Hilbert gave a construction of positive semidefinite ternary sextics with eight zeros in general position which cannot be written as a sum of squares of cubic forms. Unfortunately, the conditions on the zeros precluded any specific examples which could be easily calculated. It was not until 80 years later that R. M. Robinson relaxed the conditions in one special case, and gave an explicit form of this kind. We show that Hilbert's construction always works under these more relaxed conditions, and give many specific examples. In particular, we derive a large class of extremal psd ternary sextics with 10 zeros.

### Some Speed-Ups and Speed Limits for Real Fewnomials

We present some new algorithmic results on deciding the existence of real roots for systems of sparse polynomial equations over the real numbers. In particular:

1. We show that deciding the existence of real roots for a single $n$-variate $m$-nomial is $\mathbf{NP}$-hard for $m \geq 6n+6$ (and $n$ arbitrary).
2. A polynomial time algorithm for guessing (with success probability $1-\varepsilon$) the existence of real roots for polynomials in one variable.
Both results above are stated relative to the sparse encoding, thus circumventing the usual upper and lower complexity bounds clasically stated solely in terms of degree. We also remark that deciding the existenceof real roots for a univariate 4-nomial in (deterministic) polynomial time remains an open problem.

Result (1) is joint work with Casey Stella of Texas A&M University.

### A Refinement of the Shapiro Conjecture in Enumerative Geometry

The Shapiro conjecture asserts that all solutions to real systems of polynomial equations arising from a certain class of problems in enumerative geometry are real. It has been proven in certain cases, but is known to be false in general. However, extensive computational investigation has led to a very interesting refinement of the original conjecture.

### Trace coordinates of Teichmueller spaces of low genus

Thomas Gauglhofer, Klaus-Dieter B. Semmler

In this lecture an approach to the Teichmueller space of Fuchsian groups based on polygons and trace coordinates is described. We illustrate the simple expressions of the action of the modular group in these coordinates. They give rise to reduction algorithms on Teichmueller space.

### Genus 2 Curves that Admit a Degree 5 Map to an Elliptic Curve

We continue our study of genus 2 curves $C$ that admit a cover $C \to E$ to a genus 1 curve $E$ of prime degree $n$. These curves $C$ form an irreducible 2-dimensional subvariety ${\mathcal L}_n$ of the moduli space ${\mathcal M}_2$ of genus 2 curves. Here we study the case $n=5$. This extends earlier work for degree 2 and 3, aimed at illuminating the theory for general $n$.\par We compute a normal form for the curves in the locus ${\mathcal L}_5$ and its three distinguished subloci. Further, we compute the equation of the elliptic subcover in all cases, give a birational parameterization of the subloci of ${\mathcal L}_5$ as subvarieties of ${\mathcal M}_2$ and classify all curves in these loci which have extra automorphisms.

### Counting zeros in complex domains: average numbers and standard deviations

Bernard Shiffman, Steve Zelditch

It is well known that the average number of zeros in a domain $U$ of a degree-$N$ random polynomial (in the $SU(2)$ ensemble) equals $N$ times the (normalized spherical) area of $U$ We show that for large degree $N$, the number of zeros will very likely be close to the average. In particular, we show that when $U$ has piecewise smooth boundary, the standard deviation of the number of zeros in U is asymptotic to $\kappa N^{1/4}$. The same result holds for the zeros of random sections of powers of a positive line bundle on any compact complex curve (with the same constant $\kappa$, which is given by an explicit formula). If we pair the zero distribution by a smooth test function, then the standard deviation is instead asymptotic to $k N^{-1/2}$ (as observed by Sodin and Tsirelson for the case of polynomials). These results generalize to volumes of zero divisors on projective manifolds.

### Computing Hurwitz spectra in arithmetic form

A Riemann surface having automorphism group of order $84(g-1)$ is called a Hurwitz surface. These surfaces are intimately connected to the 2,3,7-triangle group $T$ acting by isometry on the hyperbolic plane. In recent work, a combinatorial algorithm was presented which allows the computation of the Hurwitz length spectra in floating-point form. This work has now been extended to allow exact computation of the spectra in arithmetic form. When the spectrum of $T$ is viewed in this form, a number of intriguing patterns are observed.

### Counting Bely\u\i\ Surfaces with many Automorphisms

A Bely\u\i\ surface with many automorphisms is by definition a compact Riemann surface of genus $g\geq 2$ admitting a Galois cover of the Riemann sphere branched over just three points. We shall develop an inductive method to count the number of Bely\u\i\ surfaces with many automorphisms for a fixed genus $g$ using the list of automorphism groups developed by Thomas Breuer and stored in the computer algebra package GAP.

### Topology of o-minimal Hausdorff limits and applications

Let $A$ be a bounded set of $\mathbb{R}^{n+r}$ definable in an o-minimal exapnsion $\mathcal{S}$ of the reals. Assume that for all $a \in \mathbb{R}^r,$ the fibers $A_r$ are either empty or compact. The Hausdorff limit of any sequence of fibers is known to be $\mathcal{S}$-definable. We give a formula that bounds the Betti numbers of such a limit in terms of Betti numbers of simple sets defined from the fibers of $A.$ In particular, this leads to effective upper bounds in the semialgebraic setting. In the Pfaffian setting, Gabrielov showed that the structure generated by Pfaffian functions could be described using relative closures, and the present work can be applied to give explicit upper-bounds on the Betti numbers of such relative closures.

## Schedule

Session I:   Friday January 7, 2005, 8:00 a.m.-10:50 a.m.

8:00--8:20 David Eisenbud
9:00--9:20 Charles N. Delzell
9:20--9:50 Roger Vogeler
10:00-10:20 Klaus-Dieter B. Semmler
10:30-10:50 Peter Buser

Session II:  Saturday January 8, 2005, 8:00 a.m.-10:50 a.m.

8:00--8:20 Bruce Reznick
8:30--8:50 Bonnie Sakura Huggins
9:00--9:20 David Joyner
9:30--9:50 Bernard Shiffman
10:00-10:20 Aaron D Wootton
10:30-10:50 Grigoriy Blekherman

Session III: Saturday January 8, 2005, 1:00 p.m.-5:50 p.m.

1:00--1:20 Sanjay Lall
1:30--1:50 Dmitrii V Pasechnik
2:00--2:20 Clint McCrory
2:30--2:50 Thierry Zell
3:00--3:20 James Ruffo
3:30--3:50 Maurice Rojas
4:00--4:20 Michael Greenblatt
4:30--4:50 Gou Nakamura
5:00--5:20 Jose Luis Estevez
5:30--5:50 Yasuyuki Kachi


## Dinner

We plan to organize a dinner outing for participants in the session.