Chordal networks of polynomial ideals

Transcription

1 Chordal networks of polynomial ideals Diego Cifuentes Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute of Technology Joint work with Pablo A. Parrilo (MIT) arxiv: , , Algebra Seminar - Georgia Tech Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 1 / 24

2 Polynomial systems Systems of polynomial equations have been used to model problems in areas such as: robotics, cryptography, statistics, optimization, computer vision, power networks, graph theory. Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 2 / 24

3 Polynomial systems Systems of polynomial equations have been used to model problems in areas such as: robotics, cryptography, statistics, optimization, computer vision, power networks, graph theory. Given polynomial equations F = {f 1,..., f m }, let V(F ) := {x R n : f 1 (x) = = f m (x) = } denote the associated variety. Depending on the application we might be interesting in: Feasibility Is there any solution, i.e., V(F )? Counting How many solutions? Dimension What is the dimension of V(F )? Components Decompose V(F ) into irreducible components. Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 2 / 24

4 Polynomial systems and graphs Systems coming from applications often have simple sparsity structure. We can represent this structure using graphs. Given m equations in n variables, construct a graph as: Nodes are the variables {x,..., x n 1 }. For each equation, add a clique connecting the variables appearing in that equation Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 3 / 24

5 Polynomial systems and graphs Systems coming from applications often have simple sparsity structure. We can represent this structure using graphs. Given m equations in n variables, construct a graph as: Nodes are the variables {x,..., x n 1 }. For each equation, add a clique connecting the variables appearing in that equation Example: F = {x 2 x 1 x 2 + 2x 1 + 1, x x 2, x 1 + x 2, x 2 x 3 } Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 3 / 24

6 Polynomial systems and graphs Systems coming from applications often have simple sparsity structure. We can represent this structure using graphs. Given m equations in n variables, construct a graph as: Nodes are the variables {x,..., x n 1 }. For each equation, add a clique connecting the variables appearing in that equation Example: F = {x 2 x 1 x 2 + 2x 1 + 1, x x 2, x 1 + x 2, x 2 x 3 } Question: Can the graph structure help solve polynomial systems? Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 3 / 24

7 Graphical modelling Pervasive idea in many areas, in particular: numerical linear algebra, graphical models, constraint satisfaction, database theory,... Key notions: chordality and treewidth. Many names: Arnborg, Beeri/Fagin/Maier/Yannakakis, Blair/Peyton, Bodlaender, Courcelle, Dechter, Lauritzen/Spiegelhalter, Pearl, Robertson/Seymour,... Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 4 / 24

8 Graphical modelling Pervasive idea in many areas, in particular: numerical linear algebra, graphical models, constraint satisfaction, database theory,... Key notions: chordality and treewidth. Many names: Arnborg, Beeri/Fagin/Maier/Yannakakis, Blair/Peyton, Bodlaender, Courcelle, Dechter, Lauritzen/Spiegelhalter, Pearl, Robertson/Seymour,... Remarkably (AFAIK) almost no work in computational algebraic geometry exploits this structure. Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 4 / 24

9 Graphical modelling Pervasive idea in many areas, in particular: numerical linear algebra, graphical models, constraint satisfaction, database theory,... Key notions: chordality and treewidth. Many names: Arnborg, Beeri/Fagin/Maier/Yannakakis, Blair/Peyton, Bodlaender, Courcelle, Dechter, Lauritzen/Spiegelhalter, Pearl, Robertson/Seymour,... Remarkably (AFAIK) almost no work in computational algebraic geometry exploits this structure. We hope to change this... ;) Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 4 / 24

10 Example 1: Coloring a cycle Let C n = (V, E) be the cycle graph and consider the ideal I given by the equations xi 3 1 =, i V xi 2 + x i x j + xj 2 =, ij E These equations encode the proper 3-colorings of the graph. Note that coloring the cycle graph is very easy! Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 5 / 24

11 Example 1: Coloring a cycle Let C n = (V, E) be the cycle graph and consider the ideal I given by the equations xi 3 1 =, i V xi 2 + x i x j + xj 2 =, ij E These equations encode the proper 3-colorings of the graph. Note that coloring the cycle graph is very easy! However, a Gröbner basis is not so simple: one of its 13 elements is x x 2 x 4 x 6 + x x 2 x 4 x 7 + x x 2 x 4 x 8 + x x 2 x 5 x 6 + x x 2 x 5 x 7 + x x 2 x 5 x 8 + x x 2 x 6 x 8 + x x 2 x 7 x 8 + x x 2 x x x 3 x 4 x 6 + x x 3 x 4 x 7 +x x 3 x 4 x 8 + x x 3 x 5 x 6 + x x 3 x 5 x 7 + x x 3 x 5 x 8 + x x 3 x 6 x 8 + x x 3 x 7 x 8 + x x 3 x x x 4 x 6 x 8 + x x 4 x 7 x 8 + x x 4 x2 8 + x x 5 x 6 x 8 +x x 5 x 7 x 8 + x x 5 x x x 6 x2 8 + x x 7 x2 8 + x + x 1 x 2 x 4 x 6 + x 1 x 2 x 4 x 7 + x 1 x 2 x 4 x 8 + x 1 x 2 x 5 x 6 + x 1 x 2 x 5 x 7 + x 1 x 2 x 5 x 8 +x 1 x 2 x 6 x 8 + x 1 x 2 x 7 x 8 + x 1 x 2 x x 1 x 3 x 4 x 6 + x 1 x 3 x 4 x 7 + x 1 x 3 x 4 x 8 + x 1 x 3 x 5 x 6 + x 1 x 3 x 5 x 7 + x 1 x 3 x 5 x8 + x 1 x 3 x 6 x 8 + x 1 x 3 x 7 x 8 +x 1 x 3 x x 1 x 4 x 6 x 8 + x 1 x 4 x 7 x 8 + x 1 x 4 x2 8 + x 1 x 5 x 6 x 8 + x 1 x 5 x 7 x 8 + x 1 x 5 x2 8 + x 1 x 6 x2 8 + x 1 x 7 x2 8 + x 1 + x 2 x 4 x 6 x 8 + x 2 x 4 x 7 x 8 +x 2 x 4 x x 2 x 5 x 6 x 8 + x 2 x 5 x 7 x 8 + x 2 x 5 x2 8 + x 2 x 6 x2 8 + x 2 x 7 x2 8 + x 2 + x 3 x 4 x 6 x 8 + x 3 x 4 x 7 x 8 + x 3 x 4 x2 8 + x 3 x 5 x 6 x 8 + x 3 x 5 x 7 x 8 +x 3 x 5 x x 3 x 6 x2 8 + x 3 x 7 x2 8 + x 3 + x 4 x 6 x2 8 + x 4 x 7 x2 8 + x 4 + x 5 x 6 x2 8 + x 5 x 7 x2 8 + x 5 + x 6 + x 7 + x 8 Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 5 / 24

12 Example 1: Coloring a cycle There is an alternative representation of the ideal, that respects its graphical structure. The variety can be decomposed into triangular sets: V(I) = T V(T ) where the union is overall all maximal directed paths (or chains). The number of triangular sets is 21, which is the 8-th Fibonacci number x 2 + xx8 + x 2 8 x + x 1 + x 8 x 1 x 8 x x1 2 + x1x8 + x x 2 + x 8 x x2x8 + x 2 8 x 2 + x 3 + x 8 x 2 x 8 x 3 x 8 x x3 2 + x3x8 + x x 4 + x 8 x x4x8 + x 2 8 x 4 + x 5 + x 8 x 4 x 8 x 5 x 8 x x5 2 + x5x8 + x x 6 + x 8 x 6 + x 7 + x 8 x 6 x 8 x x7x8 + x 2 8 x Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 6 / 24

13 Example 2: Ideal of adjacent minors 1 x x 3 x 1 x 2 I = {x 2i x 2i+3 x 2i+1 x 2i+2 : i < n} 23 x 2, x 3 x 2 x 5 x 3 x 4 This is the ideal of adjacent minors of the matrix ( x x 2 x 4 ) x 2n 2 x 1 x 3 x 5 x 2n x 4 x 7 x 5 x 6 x 4, x 5 x 6, x 7 x 6 x 9 x 7 x 8 x 8 x 11 x 9 x 1 x 8, x 9 The total number of irreducible components is the n-th Fibonacci number. 1,11 12,13 x 1, x 11 x 1 x 13 x 11 x 12 x 12 x 15 x 13 x 14 x 12, x 13 Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 7 / 24

14 Example 2: Ideal of adjacent minors 1 x x 3 x 1 x 2 I = {x 2i x 2i+3 x 2i+1 x 2i+2 : i < n} 23 x 2, x 3 x 2 x 5 x 3 x 4 This is the ideal of adjacent minors of the matrix ( x x 2 x 4 ) x 2n 2 x 1 x 3 x 5 x 2n x 4 x 7 x 5 x 6 x 4, x 5 x 6, x 7 x 6 x 9 x 7 x 8 x 8 x 11 x 9 x 1 x 8, x 9 The total number of irreducible components is the n-th Fibonacci number. 1,11 12,13 x 1, x 11 x 1 x 13 x 11 x 12 x 12 x 15 x 13 x 14 x 12, x 13 More generally, the ideal of adjacent minors of a k n matrix also has a simple chordal network representation. Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 7 / 24

15 Our results We introduce the notion of chordal networks, a new representation of structured polynomial ideals. An algorithm to compute chordal network representations. We show that several families of polynomial systems admit a chordal network representation of size O(n), even though the number of components is exponentially large. We show how to effectively use chordal networks to solve: feasibility, counting, dimension, elimination, radical membership and sometimes components. Implementation and experimental results. Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 8 / 24

16 Chordal graphs For a graph G, an ordering of its vertices x > x 1 > > x n 1 is a perfect elimination ordering if for each x l X l := {x m : x m is adjacent to x l, x l > x m } is a clique. A graph is chordal if it has a perfect elimination ordering. Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 9 / 24

17 Chordal graphs For a graph G, an ordering of its vertices x > x 1 > > x n 1 is a perfect elimination ordering if for each x l X l := {x m : x m is adjacent to x l, x l > x m } is a clique. A graph is chordal if it has a perfect elimination ordering. A chordal completion of G is supergraph that is chordal. Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 9 / 24

18 Elimination tree of a chordal graph The elimination tree of a graph G is the following directed spanning tree: For each l there is an arc towards its smallest neighbor p, with p > l. Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 1 / 24

19 Chordal networks A G-chordal network is a directed graph N, whose nodes are polynomial sets, satisfying the following conditions arcs follow elimination tree: if (F l, F p ) is an arc, then (l, p) is an arc of the elimination tree, where l = rank(f l ), p = rank(f p ). nodes supported on cliques: each node F of N is given a rank l := rank(f ), such that F only involves variables in the clique X l. Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

20 Chordal networks (Example) 1 2 x 3 + x 2 x7 + xx x 3 7 g(x, x6, x7) x x 2 1 x9 + x1x x 3 9 g(x1, x4, x9) x x 2 2 x5 + x2x x 3 5 g(x2, x3, x5) 3 x3 x5 x3 + x5 + x7 + x8 g(x3, x7, x8) 4 x4 x9 x4 + x5 + x8 + x9 g(x4, x8, x9) 5 g(x5, x8, x9) x5 + x7 + x8 + x9 x5 x7 x5 x9 x5 x9 6 x6 x7 g(x6, x8, x9) x6 + x7 + x8 + x9 7 x7 x9 g(x7, x8, x9) 8 x x 2 8 x9 + x8x x 3 9 g(a, b, c) := a 2 + b 2 + c 2 + ab + bc + ca 9 x Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

21 Computing chordal networks: Triangular sets Defn: A zero dimensional triangular set is T = {t,..., t n 1 } such that t = x d + g (x, x 1,..., x n 1 ), (deg x (g ) < d ). t n 2 = x d n 2 n 2 + g n 2(x n 2, x n 1 ), (deg xn 2 (g 1 ) < d n 2 ) t n 1 = g n 1 (x n 1 ) Remk: A triangular set is a Gröbner basis w.r.t. lexicographic order. Defn: Let I K[X] be a zero dimensional ideal. A triangular decomposition of I is a collection T of triangular sets, such that V(I) = T T V(T ) Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

22 Computing chordal networks (Example) The ideal I = x x 2 x 2, x 3 x, x 1 x 2, x2 2 x 2, x 2 x 3 can be decomposed into three triangular sets T 1 = (x 3 x, x 1 x 2, x 2, x 3 ), T 2 = (x 1, x 1 x 2, x 2 1, x 3 ), T 3 = (x 1, x 1 x 2, x 2 1, x 3 1). Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

23 Computing chordal networks (Example) The ideal I = x x 2 x 2, x 3 x, x 1 x 2, x2 2 x 2, x 2 x 3 can be decomposed into three triangular sets T 1 = (x 3 x, x 1 x 2, x 2, x 3 ), T 2 = (x 1, x 1 x 2, x 2 1, x 3 ), T 3 = (x 1, x 1 x 2, x 2 1, x 3 1). These triangular sets correspond to chains of a chordal network x 3 x x 1 1 x 1 x 2 2 x 2 x x 3 x 3 1 Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

24 Computing chordal networks (Example) x 3 x, xx2 x2, x 2 2 x2 x 1 x 2, x 2 2 x2 x 2 2 x2, x2x 2 3 x3 Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

25 Computing chordal networks (Example) x 3 x, xx2 x2, x 2 2 x2 x 1 x 2, x 2 2 x2 tria x 3 x, x2 x 1, x 2 1 x 1 x 2, x 2 2 x2 x 2 2 x2, x2x 2 3 x3 x 2 2 x2, x2x 2 3 x3 Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

26 Computing chordal networks (Example) x 3 x, xx2 x2, x 2 2 x2 x 1 x 2, x 2 2 x2 tria x 3 x, x2 x 1, x 2 1 x 1 x 2, x 2 2 x2 elim x 3 x x 1 x 1 x 2, x 2 2 x2 x 2 2 x2, x2x 2 3 x3 x 2 2 x2, x2x 2 3 x3 x2 2 x2, x2x 3 2 x3, x2 x 2 2 x2, x2x 3 2 x3, x2 1 Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

27 Computing chordal networks (Example) x 3 x, xx2 x2, x 2 2 x2 x 1 x 2, x 2 2 x2 tria x 3 x, x2 x 1, x 2 1 x 1 x 2, x 2 2 x2 elim x 3 x x 1 x 1 x 2, x 2 2 x2 x 2 2 x2, x2x 2 3 x3 x 2 2 x2, x2x 2 3 x3 x2 2 x2, x2x 3 2 x3, x2 x 2 2 x2, x2x 3 2 x3, x2 1 elim x 3 x x 1 x 1 x 2 x2 2 x2, x2x 3 2 x3, x2 x 2 2 x2, x2x 3 2 x3, x2 1 Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

28 Computing chordal networks (Example) x 3 x, xx2 x2, x 2 2 x2 x 1 x 2, x 2 2 x2 tria x 3 x, x2 x 1, x 2 1 x 1 x 2, x 2 2 x2 elim x 3 x x 1 x 1 x 2, x 2 2 x2 x 2 2 x2, x2x 2 3 x3 x 2 2 x2, x2x 2 3 x3 x2 2 x2, x2x 3 2 x3, x2 x 2 2 x2, x2x 3 2 x3, x2 1 x 3 x x 1 x 3 x x 1 elim x 1 x 2 x2 2 x2, x2x 3 2 x3, x2 x 2 2 x2, x2x 3 2 x3, x2 1 tria x 1 x 2 x 2, x 3 x 2 1, x 3 x 2 1, x 3 1 Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

29 Computing chordal networks (Example) x 3 x, xx2 x2, x 2 2 x2 x 1 x 2, x 2 2 x2 tria x 3 x, x2 x 1, x 2 1 x 1 x 2, x 2 2 x2 elim x 3 x x 1 x 1 x 2, x 2 2 x2 x 2 2 x2, x2x 2 3 x3 x 2 2 x2, x2x 2 3 x3 x2 2 x2, x2x 3 2 x3, x2 x 2 2 x2, x2x 3 2 x3, x2 1 x 3 x x 1 x 3 x x 1 elim x 1 x 2 x2 2 x2, x2x 3 2 x3, x2 x 2 2 x2, x2x 3 2 x3, x2 1 tria x 1 x 2 x 2, x 3 x 2 1, x 3 x 2 1, x 3 1 x 3 x x 1 elim x 1 x 2 x 2 x 2 1 x 2 1 x 3 x 3 x 3 1 Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

30 Computing chordal networks (Example) x 3 x, xx2 x2, x 2 2 x2 x 1 x 2, x 2 2 x2 tria x 3 x, x2 x 1, x 2 1 x 1 x 2, x 2 2 x2 elim x 3 x x 1 x 1 x 2, x 2 2 x2 x 2 2 x2, x2x 2 3 x3 x 2 2 x2, x2x 2 3 x3 x2 2 x2, x2x 3 2 x3, x2 x 2 2 x2, x2x 3 2 x3, x2 1 x 3 x x 1 x 3 x x 1 elim x 1 x 2 x2 2 x2, x2x 3 2 x3, x2 x 2 2 x2, x2x 3 2 x3, x2 1 tria x 1 x 2 x 2, x 3 x 2 1, x 3 x 2 1, x 3 1 x 3 x x 1 x 3 x x 1 elim x 1 x 2 x 2 x 2 1 x 2 1 merge 1 2 x 1 x 2 x 2 x 2 1 x 3 x 3 x x 3 x 3 1 Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

31 Main results Thm 1: Chordal triangularization obtains a G-chordal network, whose chains give a triangular decomposition of F. Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

32 Main results Thm 1: Chordal triangularization obtains a G-chordal network, whose chains give a triangular decomposition of F. For nice cases the chordal network obtained has linear size. Thm 2: Let F be a family of structured polynomial systems such that V(F K[X l ]) is bounded for any F F and for any maximal clique X l. Then any F F admits a chordal network representation of size O(n). Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

33 Chordal networks in computational algebra Given a triangular chordal network N of an ideal I, we can compute in linear time: the cardinality of V(I). the dimension of V(I) the top dimensional part of V(I). We also show efficient algorithms for: radical ideal membership. computing equidimensional (sometimes irreducible) components. Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

34 Chordal networks in computational algebra Given a triangular chordal network N of an ideal I, we can compute in linear time: the cardinality of V(I). the dimension of V(I) the top dimensional part of V(I). We also show efficient algorithms for: radical ideal membership. computing equidimensional (sometimes irreducible) components. The main difficulty is that there might be exponentially many chains. It can be overcomed by cleverly using dynamic programming (or message-passing). Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

35 Radical ideal membership Problem: Determine if a polynomial h vanishes on a variety. Given a chordal network, we need to determine whether h mod C for all chains C Note that there might be exponentially many chains. Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

36 Radical ideal membership (Sketch) h(x) h(x) x 3 + x 2 x7 + xx x 3 7 x 2 + xx6 + xx7 + x x6x7 + x h a 1 h b 1 h b x 6 x 7 x x6x8 + x6x9 + x x8x9 + x 2 9 x 6 + x 7 + x 8 + x 9 1 h b ha 6 x 7 x 9 x x7x8 + x7x9 + x x8x9 + x ha 7 x x 2 8 x9 + x8x x hb hc 6 1 h 8 9 x h 9 = Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

37 Implementation and examples Implemented in Sage. Graph colorings (counting q-colorings) Cryptography ( baby AES, Cid et al.) Sensor Network localization Discretization of differential equations Algebraic statistics Reachability in vector addition systems Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT 17 2 / 24

38 Application: Vector addition systems Given a set of vectors B Z n, construct a graph with vertex set N n in which u, v N n are adjacent if u v ±B. The reachability problem is to describe the connected components of the graph Problem: There are n card decks on a circle. Given any four consecutive decks we can: take one card from the inner decks and place them in the outer decks. take one card from the outer decks and place them on the inner decks. Initially the number of cards in the decks are 1, 2,..., n. Is it possible to reverse the number of cards in the decks? Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

39 Application: Vector addition systems The problem is equivalent to determining whether f n I n, where f n := x x 2 1 x 3 2 x n n 1 x n x n 1 1 x n 1, I n := {x i x i+3 x i+1 x i+2 : i < n}, and where the indices are taken modulo n. We compare our radical membership test with Singular (Gröbner bases) and Epsilon (triangular decompositions). n ChordalNet Singular Epsilon Test result true false false false true false true false false false true Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

40 Application: Algebraic statistics (Evans et al.) Consider the binomial ideal I n,n 2 that models a 2D dimensional generalization of the birth-death Markov process. We fix n 2 = 1. We can compute faster all irreducible components of the ideal than the Macaulay2 package Binomials. n #components ChordTriaD :: ::1 ::4 ::13 :2:1 :37:35 12:22:19 time Binomials :: :: ::1 ::12 :3: 4:15:36 - Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

41 Application: Algebraic statistics (Evans et al.) Consider the binomial ideal I n,n 2 that models a 2D dimensional generalization of the birth-death Markov process. We fix n 2 = 1. We can compute faster all irreducible components of the ideal than the Macaulay2 package Binomials. n #components ChordTriaD :: ::1 ::4 ::13 :2:1 :37:35 12:22:19 time Binomials :: :: ::1 ::12 :3: 4:15:36 - Our methods are particularly efficient to compute the highest dimensional components. Highest 5 dimensions Highest 7 dimensions n #comps time :1:7 :4:54 :15:12 :41:52 1:34:5 :5:2 :41:41 3:3:29 9:53:9 Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

42 Summary Chordal structure can notably help in computational algebraic geometry. Under assumptions (chordality + algebraic structure), tractable! Yields practical, implementable algorithms. A Macaulay2 package Chordal is in preparation. Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24

43 Summary Chordal structure can notably help in computational algebraic geometry. Under assumptions (chordality + algebraic structure), tractable! Yields practical, implementable algorithms. A Macaulay2 package Chordal is in preparation. If you want to know more: D. Cifuentes, P.A. Parrilo (217), Chordal networks of polynomial ideals. SIAM J. of Applied Algebra and Geometry, 1(1): arxiv: D. Cifuentes, P.A. Parrilo (216), Exploiting chordal structure in polynomial ideals: a Gröbner basis approach. SIAM J. Discrete Math., 3(3): arxiv: D. Cifuentes, P.A. Parrilo (216), An efficient tree decomposition method for permanents and mixed discriminants. Linear Alg. and its Appl., 493: arxiv: Thanks for your attention! Cifuentes (MIT) Chordal networks of polynomial ideals Algebra seminar GT / 24