Challenges in Computational Algebraic Geometry Challenges in Computational Algebraic David A. Cox Geometry Challenge 1: Other Computational Algebraic Geometry Workshop Disciplines Article in Nature The Mathematics Methods Challenge 2: David A. Cox The Range of Computations Resultants A Joint Paper Department of Mathematics and Computer Science Amherst College Challenge 3: Loving Bad dac@cs.amherst.edu Algorithms Factoring over the FoCM’08 Rationals Factoring over Number Fields 20 June 2008 Sudoku
Challenges Challenges in Computational Algebraic Geometry There are many challenges facing Computational Algebraic David A. Cox Geometry: Challenge 1: Practical: Do big problems using existing algorithms Other Disciplines and hardware. Article in Nature The Mathematics Theoretical: Find better algorithms. Also understand Methods the complexity of existing algorithms. Challenge 2: The Range of Computations Resultants A Joint Paper This Talk Challenge 3: Loving Bad I will discuss some completely different challenges facing Algorithms Factoring over the Computational Algebraic Geometry. Rationals Factoring over Number Fields Sudoku
Outline Challenges in Computational Algebraic Challenge 1: Other Disciplines 1 Geometry Article in Nature David A. Cox The Mathematics Challenge 1: Methods Other Disciplines Article in Nature The Mathematics Challenge 2: The Range of Computations 2 Methods Resultants Challenge 2: The Range of A Joint Paper Computations Resultants A Joint Paper Challenge 3: Loving Bad Algorithms 3 Challenge 3: Loving Bad Factoring over the Rationals Algorithms Factoring over the Factoring over Number Fields Rationals Factoring over Sudoku Number Fields Sudoku
Nature, 20 January 2000 Challenges in Computational A Synthetic Oscillatory Network of Transcriptional Regulators Algebraic Geometry Michael B. Elowitz & Stanislas Leibler David A. Cox Challenge 1: Other Departments of Molecular Biology and Physics, Princeton Disciplines Article in Nature The Mathematics Networks of interacting biomolecules carry out many Methods essential functions in living cells, but the ‘design principles’ Challenge 2: The Range of underlying the functioning of such intracellular networks Computations Resultants remain poorly understood. A Joint Paper Challenge 3: Loving Bad Here we present the design and construction of a synthetic Algorithms Factoring over the network to implement a particular function. We used three Rationals Factoring over transcriptional repressor systems to build an oscillating Number Fields Sudoku network, termed the repressilator, in Escherichia coli .
From “Box 1” of the Article Challenges in Three repressor-protein concentrations p i and their Computational Algebraic corresponding mRNA concentrations m i ( i is lacI , tetR , cI ) Geometry are treated as continuous dynamical variables. David A. Cox Challenge 1: The kinetics of the system are determined by six coupled Other Disciplines first-order differential equations: Article in Nature The Mathematics α Methods dm i + α 0 = − m i + Challenge 2: 1 + p n dt The Range of j Computations Resultants dp i − β ( p i − m i ) A Joint Paper = dt Challenge 3: Loving Bad Algorithms for i = lacI , tetR , cI , j = cI , lacI , tetR and n , α , α 0 , β > 0. Factoring over the Rationals Factoring over Number Fields Question Sudoku What are the steady-state solutions?
Steady State Solutions Challenges in Computational Algebraic The steady state solutions are solutions of the system: Geometry α David A. Cox + α 0 0 = − m i + 1 + p n Challenge 1: j Other Disciplines − β ( p i − m i ) 0 = Article in Nature The Mathematics Methods Write the indices as i = 1 , 2 , 3, j = 2 , 3 , 1. Challenge 2: The Range of Computations The System of Equations Resultants A Joint Paper α 2 + α 0 0 = − p 1 + Challenge 3: 1 + p n Loving Bad α 3 + α 0 0 = − p 2 + Algorithms 1 + p n Factoring over the α 1 + α 0 = − p 3 + Rationals 0 1 + p n Factoring over Number Fields Sudoku
Real Solutions Challenges in Computational Claim Algebraic Geometry Assume α , α 0 > 0. David A. Cox The equation α Challenge 1: 1 + p n + α 0 Other p = Disciplines Article in Nature The Mathematics has a unique real solution, denoted p . Methods Challenge 2: The unique real solution of The Range of Computations Resultants α 2 + α 0 0 = − p 1 + A Joint Paper 1 + p n α Challenge 3: 3 + α 0 = − p 2 + 0 Loving Bad 1 + p n Algorithms α 1 + α 0 0 = − p 3 + Factoring over the 1 + p n Rationals Factoring over Number Fields is given by p 1 = p 2 = p 3 = p . Sudoku
Proof for n = 2 Set p = p 1 , a = α , b = α 0 and eliminate p 2 , p 3 : Challenges in Computational Algebraic ( p 3 − bp 2 + p − a − b )( 1 + 2 a 2 + a 4 + 5 ab + 4 a 3 b + 3 b 2 + Geometry 8 a 2 b 2 + 8 ab 3 + a 3 b 3 + 3 b 4 + 3 a 2 b 4 + 3 ab 5 + b 6 − ap − David A. Cox 2 a 2 bp − 2 ab 2 p − a 3 b 2 p − 2 a 2 b 3 p − ab 4 p + 3 p 2 + 4 a 2 p 2 + Challenge 1: Other 12 abp 2 + 3 a 3 bp 2 + 9 b 2 p 2 + 12 a 2 b 2 p 2 + 18 ab 3 p 2 + 9 b 4 p 2 + Disciplines 3 a 2 b 4 p 2 + 6 ab 5 p 2 + 3 b 6 p 2 − 2 ap 3 + a 3 p 3 − 2 a 2 bp 3 − Article in Nature The Mathematics 4 ab 2 p 3 − 2 a 2 b 3 p 3 − 2 ab 4 p 3 + 3 p 4 + 3 a 2 p 4 + 9 abp 4 + 9 b 2 p 4 + Methods Challenge 2: 4 a 2 b 2 p 4 + 12 ab 3 p 4 + 9 b 4 p 4 + 3 ab 5 p 4 + 3 b 6 p 4 − ap 5 − The Range of Computations 2 ab 2 p 5 − ab 4 p 5 + p 6 + a 2 p 6 + 2 abp 6 + 3 b 2 p 6 + 2 ab 3 p 6 + Resultants A Joint Paper 3 b 4 p 6 + b 6 p 6 ) = 0 Challenge 3: Loving Bad Algorithms The second factor is a polynomial H of degree 6 in p . Factoring over the Rationals Factoring over Number Fields Small Discriminant Calculation Sudoku p 3 − bp 2 + p − a − b has a unique real root when a , b > 0.
Proof for n = 2, Continued Challenges in Computational Larger Discriminant Calculation Algebraic Geometry The polynomial H is positive when p , a , b > 0. David A. Cox Challenge 1: Other (Suggested by Fabrice Rouillier) The discriminant of H is Disciplines Article in Nature Disc ( H , p ) = a 16 ( a 2 + b 6 + 3 b 2 + 3 b 4 + 1 + 2 ab + 2 ab 3 ) P The Mathematics Methods Challenge 2: The Range of where P is a sum (no subtractions) of monomials in a , b with Computations Resultants a constant term 16384. The leading coefficient of H A Joint Paper Challenge 3: a 2 + b 6 + 3 b 2 + 3 b 4 + 1 + 2 ab + 2 ab 3 Loving Bad Algorithms Factoring over the Rationals is strictly positive when a , b > 0, so H has no root at infinity. Factoring over Number Fields Sudoku So the number of real roots of H is constant when a , b > 0 . QED
Better Proof for all n > 0 Challenges in Computational (Suggested by André Galligo) Algebraic Geometry Key Point David A. Cox α If α , α 0 , n > 0, then p �→ 1 + p n + α 0 is strictly decreasing. Challenge 1: Other Disciplines Article in Nature Assume The Mathematics α 2 + α 0 = p 1 Methods 1 + p n α Challenge 2: 3 + α 0 p 2 = The Range of 1 + p n Computations α 1 + α 0 p 3 = Resultants 1 + p n A Joint Paper Challenge 3: and suppose for example p 2 < p 3 . Then Loving Bad Algorithms Factoring over the p 2 < p 3 ⇒ p 1 > p 2 ⇒ p 3 < p 1 ⇒ p 2 > p 3 , Rationals Factoring over Number Fields Sudoku a contradiction. QED
From the “Methods” Section of the Article Challenges in Computational Algebraic Time-lapse microscopy was conducted on a Zeiss Axiovert Geometry 135TV microscope equipped with a 512 × 512-pixel cooled David A. Cox CCD camera (Princeton Instruments). Challenge 1: Other Disciplines Bright-field (0.1 s) and epifluorescence (0.05–0.5 s) Article in Nature The Mathematics exposures were taken periodically (every 5 or 10 min). All Methods light sources (standard 100 W Hg and halogen lamps) were Challenge 2: The Range of shuttered between exposures. Computations Resultants A Joint Paper A fast Fourier transform was applied to the temporal Challenge 3: Loving Bad fluorescence signal from each analyzed cell lineage and Algorithms Factoring over the divided by the transform of a decaying exponential with a Rationals Factoring over time constant of 90 min, the measured lifetime of GFPaav. Number Fields Sudoku
More from “Box 1” of the Article The system of differential equations has a unique steady Challenges in Computational state, which becomes unstable when Algebraic Geometry ( β + 1 ) 2 David A. Cox 3 X 2 < β 4 + 2 X Challenge 1: Other Disciplines where Article in Nature X = α np n − 1 The Mathematics Methods ( 1 + p n ) 2 Challenge 2: The Range of Computations and p is the solution to Resultants A Joint Paper α Challenge 3: 1 + p n + α 0 p = Loving Bad Algorithms Factoring over the Rationals Factoring over Number Fields No Justification Whatsoever! Sudoku This is all they say about uniqueness!
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