This paper presents a radically new approach to the century old problem of computing the implicit equation of a parametric surface. For surfaces without base points, the new method expresses the implicit equation in a determinant which is one fourth the size of the conventional expression based on Dixon's resultant. If base points do exist, previous implicitization methods either fail or become much more complicated, while the new method actually simplifies.

In this paper we conduct a careful study of the equivalence classes of ternary cubics under general complex linear changes of variables. Our new results are based on the method of moving frames and involve triangular decompositions of algebraic varieties. We provide a computationally efficient algorithm that matches an arbitrary ternary cubic with its canonical form and explicitly computes a corresponding linear change of coordinates. We also describe a classification of the symmetry groups of ternary cubics.

We give an introduction to programming methods, software systems, and algorithms, suitable for parallelizing Computer Algebra on modern multiprocessor workstations. As concrete examples we present multi-threaded programming and its use in the PARSAC-2 system for parallel symbolic computation, and we present some examples of parallel algorithms useful for solving systems of polynomial equations.

Contents Preface 2 Acknowledgments ............................. 4 0.1 g-grading ................................. 4 0.2 Monomial matrices ............................ 5 0.3 Complexes and resolutions ........................ 6 0.5 Simplicial complexes and hornology ................... 8 0.6 Irreducible decomposition ........................ 10 Lecture I: Squarefree monomial ideals 11 1.1 Equivalent descriptions .......................... 11 1.3 Free resolutions .............................. 14 Lecture II: Borel-fixed monomial ideals 17 2.1 Group actions ............................... 17 2.2 Generic initial ideals ........................... 18 2.3 The Eliahou-Kervaire resolution ..................... 18 2.4 Lex-segment ideals ............................ 21 3.1 Monomial ideals in two variables .................... 22 3.2 Buchberger's second criterion ...................... 23 3.3 Resolution "by picture" ......................... 24 3.4 Planar graphs .............................

We give a variant of the homogeneous Buchberger algorithm for positively graded lattice ideals. Using this algorithm we solve the Sullivant computational commutative algebra challenge 1. 1

88 The hull resolution is not minimal: hull(ML) is obtained from the cell complex above by introducing three short edges into each large "triangle", leaving only hexagons and small triangles. The hull resolution can be made minimal by removing any one of the three edges from each resulting small "down " triangle, but we have chosen the method in the diagram for aesthetic reasons. Solution B.4.2 The hull complex of ML consists of shifts of the up and down triangles below.PSfrag replacements

My goal in this chapter of lecture notes is to introduce the basic notions of a statistical model and an algebraic variety and show how they are related to one another. The title “Statistical Models are Algebraic Varieties” could be considered as a mantra of algebraic statistics. It is the ubiquity of the polynomial structures that arise when describing statistical models that makes some statistical problems amenable to algebraic techniques. The algebraic perspective on statistical models provides a number of interesting applications, which I will discuss in more detail in later chapters. The goal of this chapter is only to show that it is not so far fetched to think that algebra could be useful in statistics by illustrating that a number of elementary statistical models have the structure of an algebraic variety. On a more pragmatic note, this chapter will also serve as a primary place to introduce much of the terminology and notation that will be used throughout the rest of the lecture notes. But what is algebraic statistics? I am sure that the answer will vary greatly depending on who is asked. In the broadest possible sense, algebraic statistics is merely the study of the algebraic structures underlying statistical inference. Which algebraic structures are emphasized will vary greatly from person to person. The focus of my research in this area has been on the ways that algebraic geometry and the ensuing commutative algebra and combinatorics arise in statistical inference, and so I will tend to focus on these aspects in these lecture notes. Of course, one cannot talk about everything so I will have to leave out lots of interesting results. Why study algebraic statistics? Again, the answer will vary greatly depending on who is asked. Algebraic statistics is certainly not a magic bullet to solve all statistical problems. For me, the approach has always been to merely try to understand a statistical problem by looking through algebraic glasses. At the end of the day, this new point of view may lead to new algebraic solutions to statistical problems. It might explain unusual phenomena that statisticians observe. It might lead to purely mathematical problems, interesting in their own right, that are not informative for statistics. It might lead to a unifying framework for discussing and exploring many related statistical problems.

gerVerlag, New York, 1995. 96 [13] D. Eisenbud, M. Mustata, and M. Stillman. Cohomology of sheaves on toric varieties. Preprint, 1998. [14] S. Eliahou and M. Kervaire. Minimal resolutions of some monomial ideals. J. Algebra, 129(1):1-25, 1990. [15] R. Gebauer and H. M. Moller. On an installation of Buchberger's algorithm. J. Symbolic Comput., 6(2-3):275-286, 1988. Computational aspects of commutative algebra. [16] S. Goto and K. Watanabe. On graded rings, II (Z n -graded rings). Tokyo J. Math., 1(2):237-261, 1978. [17] H.-G. Grabe. The canonical module of a Stanley-Reisner ring. J. Algebra, 86:272{ 281, 1984. [18] D. Grayson and M. Stillman. Macaulay 2. Available by at the website http://www.math.uiuc.edu/Macaulay2/. A software system for algebraic geometry and commutative algebra. [19] M. Hochster. Cohen-Macaulay rings, combinatorics, and simplicial complexes. I

F13.54> ! is spanned as a k-vector subspace of k(t; t 1 ) by the monomials t 2 ; t 1 ; t; t 2 ; t 3 ; : : : (with t 0 = 1 missing). These powers of t are the inverses of the powers of t that aren't in k[t 3 ; t 4 ; t 5 ]; see [7, Corollary 4.3.8]. Therefore, ! is the image under the functor of the monomial module M ! which is k-spanned by the inverses of the Laurent monomials not in M L , where L = ker([4 3 5]). What does this have to do with Alexander duality? A picture of the monomial module M ! is obtained from the picture of

In this paper we present algorithmic and complexity results for polynomial sign evaluation over two real algebraic numbers, and for real solving of bivariate polynomial systems. Our main tool is signed polynomial remainder sequences; we exploit recent advances in univariate root isolation as well as multipoint evaluation techniques.