CS6842 - Algorithmic Algebra

Meetings  

• Theme 1 : Introduction, Motivation and The Language - 2 meetings

  • Meeting 01 : Wed, Jul 31, 02:00 pm-04:00 pm

  Administrative announcements <br> The basis objects - Polynomials. <br> Why study them? And what are some algorithmic challenges? <br>Three representative applications: <ul> <li>From Graph Algorithms : Testing for perfect matching.</li> <li>From Number theory : Testing Primality - Agrawal-Biswas Formulation.</li> <li>From Robotics : Robot Motion Planning.</li> </ul> Overview of the course.

  References	<ul> <li>For Tutte's theorem - <a href="http://www.stanford.edu/class/cme305/Lectures/Lecture8.pdf‎">Notes</a> - Section 2 and 3.</li> <li>For the primality testing application - <a href="http://www.cse.iitk.ac.in/users/manindra/algebra/identity.pdf">the original paper by Agrawal-Biswas</a>. <li>For the robot motion planning - section 2,3 and 4 of the <a href="http://www-math.mit.edu/phase2/UJM/vol1/GRAYSO~2.PDF">article in MIT undergraduate journal of mathematics</a>. </ul>
  Exercises
  Reading	<ul> <li>For Tutte's theorem for general graphs - <a href="http://www.stanford.edu/class/cme305/Lectures/Lecture8.pdf‎">Notes</a> - details in section 3.</li> </ul>

  Administrative announcements
  The basis objects - Polynomials.
  Why study them? And what are some algorithmic challenges?
  Three representative applications:

  • From Graph Algorithms : Testing for perfect matching.
  • From Number theory : Testing Primality - Agrawal-Biswas Formulation.
  • From Robotics : Robot Motion Planning.

  Overview of the course.
  

  References	:	For Tutte's theorem - Notes - Section 2 and 3. For the primality testing application - the original paper by Agrawal-Biswas. For the robot motion planning - section 2,3 and 4 of the article in MIT undergraduate journal of mathematics.
  Reading	:	For Tutte's theorem for general graphs - Notes - details in section 3.

  • Meeting 02 : Fri, Aug 02, 02:00 pm-04:00 pm

  Another example application - geometric theorem proving. <br>Solving the system of polynomial equations - the informal view, ideals, Groebner basis, elimination. <br> The formal notations and language : Groups, Rings, Fields - the basic theorems.

  References	<ul> <li>[Mishra] Chapter 1, Chapter 2</li> </ul>
  Exercises	<ul> <li>Prove that cosets cannot intersect within themselves. Identify the identity element of the quotient group.</li> <li>Prove that Z_n is a field if and only of n is a prime number.</li> <li></li> </ul>
  Reading

  Another example application - geometric theorem proving.
  Solving the system of polynomial equations - the informal view, ideals, Groebner basis, elimination.
  The formal notations and language : Groups, Rings, Fields - the basic theorems.
  

  References	:	[Mishra] Chapter 1, Chapter 2
  Exercises	:	Prove that cosets cannot intersect within themselves. Identify the identity element of the quotient group. Prove that Z_n is a field if and only of n is a prime number.

• Theme 2 : Ideals, Grobner Basis and Algorithms - 9 meetings

  • Meeting 03 : Tue, Aug 06, 02:00 pm-04:00 pm

  Ideals, Examples. Principal Ideal. Polynomial Ideals. <br> Polynomial Rings in one variable. Ideal generated by polynomials. Leading terms, polynomial division algorithm. All Ideals of F[x] are principal ideals. Non-algorithmic proof. Need for an algorithm.

  References	<ul> <li>[CLO] Chapter 1, Section 5.</li> </ul>
  Exercises
  Reading

  Ideals, Examples. Principal Ideal. Polynomial Ideals.
  Polynomial Rings in one variable. Ideal generated by polynomials. Leading terms, polynomial division algorithm. All Ideals of F[x] are principal ideals. Non-algorithmic proof. Need for an algorithm.
  

  References

  :

  [CLO] Chapter 1, Section 5.

  • Meeting 04 : Wed, Aug 07, 02:00 pm-04:00 pm

  GCD and the generator of the ideal. <br> Euclidean Algorithm - termination and correctness in the language of ideals. Answering Ideal Membership problem for univariate polynomial rings. <br><br> Monomials, Ideals generated by monomials. <br>Dickson's Lemma - Every Monomial ideal if finitely generated.

  References	<ul> <li>[Mishra] : Section 2.2</li> </ul>
  Exercises	We did a "graphic" proof of Dickson's Lemma. Write down the complete inductive proof of Dickson's Lemma.
  Reading	<ul> <li>[CLO] Chapter 2, Section 4</li> </ul>

  GCD and the generator of the ideal.
  Euclidean Algorithm - termination and correctness in the language of ideals. Answering Ideal Membership problem for univariate polynomial rings.
  
  Monomials, Ideals generated by monomials.
  Dickson's Lemma - Every Monomial ideal if finitely generated.
  

  References	:	[Mishra] : Section 2.2
  Exercises	:	We did a "graphic" proof of Dickson's Lemma. Write down the complete inductive proof of Dickson's Lemma.
  Reading	:	[CLO] Chapter 2, Section 4

  • Meeting 05 : Tue, Aug 13, 02:00 pm-04:00 pm

  Division in multivariate polynomials. Ordering of monomials. Lex ordering and graded lex ordering. Multivariate multi-polynomial division algorithm - Examples and arriving at the definition of a reminder. Final algorithm and its correctness. Uniqueness of the reminder.

  References	<ul> <li>[CLO] Chapter 2, Section 4 and 5.</li> </ul>
  Exercises
  Reading

  Division in multivariate polynomials. Ordering of monomials. Lex ordering and graded lex ordering. Multivariate multi-polynomial division algorithm - Examples and arriving at the definition of a reminder. Final algorithm and its correctness. Uniqueness of the reminder.
  

  References

  :

  [CLO] Chapter 2, Section 4 and 5.

  • Meeting 06 : Tue, Aug 20, 02:00 pm-04:00 pm

  Attempting to solve the ideal membership problem using division algorithm - the mindblocks because of the order of the divisors. <br> Leading Terms and Initial Ideals. From Dickson's Lemma to Hilbert Basis theorem. The "Nice Basis" in the proof. Non-constructive nature. <br>Using the Groebner condition to get across the mindblocks in the attempt on ideal membership problem.

  References	<ul> <li>[CLO] Chapter 2, Section 5 and 6.</li> </ul>
  Exercises
  Reading

  Attempting to solve the ideal membership problem using division algorithm - the mindblocks because of the order of the divisors.
  Leading Terms and Initial Ideals. From Dickson's Lemma to Hilbert Basis theorem. The "Nice Basis" in the proof. Non-constructive nature.
  Using the Groebner condition to get across the mindblocks in the attempt on ideal membership problem.
  

  References

  :

  [CLO] Chapter 2, Section 5 and 6.

  • Meeting 07 : Wed, Aug 21, 02:00 pm-04:00 pm

  Computing Grobner basis. Constructing violating examples for Grobner condition. Leading the intuition to S-polynomials and Buchberger's criterion. Buchberger's algorithm. Termination and Correctness. Proof of Buchberger's criterion - Main lemma : any violating example can be expressed in terms of S-polynomials.

  References	<ul> <li>[CLO] Chapter 2, Section 5 and 6.</li> <li><a href="http://www.kent.ac.uk/smsas/personal/gdb/MA574/week4.pdf">Hilbert Basis Theorem and Grobner Basis - Notes by Gavin Brown</a></li> </ul>
  Exercises
  Reading

  Computing Grobner basis. Constructing violating examples for Grobner condition. Leading the intuition to S-polynomials and Buchberger's criterion. Buchberger's algorithm. Termination and Correctness. Proof of Buchberger's criterion - Main lemma : any violating example can be expressed in terms of S-polynomials.
  

  References

  :

  [CLO] Chapter 2, Section 5 and 6. Hilbert Basis Theorem and Grobner Basis - Notes by Gavin Brown

  • Meeting 08 : Sat, Aug 24, 10:00 am-11:30 am

  Proof of Buchberger's criterion. <br> Review.

  References	<ul> <li>[CLO] Chapter 2, Section 5 and 6.</li> <li><a href="http://www.kent.ac.uk/smsas/personal/gdb/MA574/week4.pdf">Hilbert Basis Theorem and Grobner Basis - Notes by Gavin Brown</a></li> </ul>
  Exercises
  Reading

  Proof of Buchberger's criterion.
  Review.
  

  References

  :

  [CLO] Chapter 2, Section 5 and 6. Hilbert Basis Theorem and Grobner Basis - Notes by Gavin Brown

  • Meeting 09 : Tue, Aug 27, 02:00 pm-04:00 pm

  Minimal and Reduced Grobner Basis. Ideal Equality Problem. Uniqueness of Reduced Grobner basis. Elimination Theory and Elimination Ideals. Grobner Basis for the Elimination Ideals. Relating to the Robotics Arms problem.

  References	<ul> <li>[CLO] Chapter 2, Section 5 and 6.</li> </ul>
  Exercises
  Reading

  Minimal and Reduced Grobner Basis. Ideal Equality Problem. Uniqueness of Reduced Grobner basis. Elimination Theory and Elimination Ideals. Grobner Basis for the Elimination Ideals. Relating to the Robotics Arms problem.
  

  References

  :

  [CLO] Chapter 2, Section 5 and 6.

  • Meeting 10 : Wed, Aug 28, 02:00 pm-04:00 pm

  Applications of Grobner Basis. 3-coloring via Grobner basis. Testing membership in Kernel and Image of Ring Homomorphisms. Applications to Integer Programming. Structure of the pre-images.

  References	<ul> <li><a href="http://documents.kenyon.edu/math/CWendler.pdf">Chapter 4 of this link.</a></li> </ul>
  Exercises
  Reading

  Applications of Grobner Basis. 3-coloring via Grobner basis. Testing membership in Kernel and Image of Ring Homomorphisms. Applications to Integer Programming. Structure of the pre-images.
  

  References

  :

  Chapter 4 of this link.

  • Meeting 11 : Sat, Aug 31, 11:05 am-12:00 pm

  <a>Shorter Extra Lecture after Quiz-1</a><br>Proof of the characterization of the Image of the k-algebra homomorphism. Observation about the Grobner basis in the special case of integer programming. Defining the monomial ordering to bring in optimization. Proof of optimality of the solution.

  References	<ul> <li><a href="http://documents.kenyon.edu/math/CWendler.pdf">Chapter 4 of this link.</a></li> </ul>
  Exercises
  Reading

  Shorter Extra Lecture after Quiz-1
  Proof of the characterization of the Image of the k-algebra homomorphism. Observation about the Grobner basis in the special case of integer programming. Defining the monomial ordering to bring in optimization. Proof of optimality of the solution.
  

  References

  :

  Chapter 4 of this link.

• Theme 3 : Algorithms for Factorization of Polynomials - 18 meetings

  • Meeting 12 : Tue, Sep 03, 02:15 pm-03:05 pm

  <a color="blue">Shorter Lecture (compensating for the lecture)</a> <br> From Root finding to factorization of polynomials. Why or when is a polynomial completely factorizable over the underlying ring/field? Why should they be unique factorizable? Informal answers, and directions to explore.

  References
  Exercises
  Reading

  Shorter Lecture (compensating for the lecture)
  From Root finding to factorization of polynomials. Why or when is a polynomial completely factorizable over the underlying ring/field? Why should they be unique factorizable? Informal answers, and directions to explore.
  

  References

  :

  None

  • Meeting 13 : Wed, Sep 04, 02:00 pm-04:00 pm

  Irreducible and Prime Elements in an Integral Domain. Primes are irreducible. When is it that all irreducibles are primes? A proof that this is exactly when the domain is a UFD.<br> A field is a UFD. Every principal Ideal Domain (example: F[x] and Z) is a UFD.<br> What about rings like F[x1,x2], and Z[x]? <br> Gauss's theorem : If R is a UFD, then so is R[x]. <br> Proof of the theorem using the characterization about irreducibles in R[x]. The case when the irreducibles are from R itself (Gauss's Lemma).

  References	<ul> <li><a href="http://math.harvard.edu/~waffle/ufds2.pdf">Notes by Waffle.</a></li> </ul>
  Exercises
  Reading

  Irreducible and Prime Elements in an Integral Domain. Primes are irreducible. When is it that all irreducibles are primes? A proof that this is exactly when the domain is a UFD.
  A field is a UFD. Every principal Ideal Domain (example: F[x] and Z) is a UFD.
  What about rings like F[x1,x2], and Z[x]?
  Gauss's theorem : If R is a UFD, then so is R[x].
  Proof of the theorem using the characterization about irreducibles in R[x]. The case when the irreducibles are from R itself (Gauss's Lemma).
  

  References

  :

  Notes by Waffle.

  • Meeting 14 : Tue, Sep 10, 02:00 pm-04:00 pm

  Continuing the proof of Gauss's theorem : The case when the irreducibles are from the R[x]. Moving the argument to the Field of Fractions.<br> Conclusion : Two tasks are well-framed.<br> <ul> <li>How do we detect irreducibility?</li> <li>How do we factorize into irreducibile factors?</li> </ul> Eisenstein criterion for irreducibility.

  References	<ul> <li></li> </ul>
  Exercises
  Reading

  Continuing the proof of Gauss's theorem : The case when the irreducibles are from the R[x]. Moving the argument to the Field of Fractions.
  Conclusion : Two tasks are well-framed.
  

  • How do we detect irreducibility?
  • How do we factorize into irreducibile factors?

  Eisenstein criterion for irreducibility.
  

  References

  :

  • Meeting 15 : Wed, Sep 11, 02:00 pm-04:00 pm

  Quotient Rings. Irreducibility and Quotient Rings. First Isomorphism Theorem for the Quotient Ring. <br> Application 1 : Chinese remaindering theorem.<br> Application 2 : From Quotient Rings of Irreducibile polynomials to Field extensions.

  References	<ul> <li>Some of the parts are adapted from <a href="http://www.cmi.ac.in/~ramprasad/lecturenotes/comp_numb_theory/lecture0506.pdf">here</a></li> <li><a href="http://sierra.nmsu.edu/morandi/OldWebPages/Math331Spring2003/Chapter5.pdf">More on Quotient Rings</a></li> </ul>
  Exercises
  Reading

  Quotient Rings. Irreducibility and Quotient Rings. First Isomorphism Theorem for the Quotient Ring.
  Application 1 : Chinese remaindering theorem.
  Application 2 : From Quotient Rings of Irreducibile polynomials to Field extensions.
  

  References

  :

  Some of the parts are adapted from here More on Quotient Rings

  • Meeting 16 : Tue, Sep 17, 02:00 pm-04:00 pm

  Quick introduction to vector spaces. Viewing field extensions as vector spaces. Characterestic of a field. Sizes of fields. Constructing extensions and uniqueness of fields of a given size (up to isomorphism).

  References
  Exercises
  Reading

  Quick introduction to vector spaces. Viewing field extensions as vector spaces. Characterestic of a field. Sizes of fields. Constructing extensions and uniqueness of fields of a given size (up to isomorphism).
  

  References

  :

  None

  • Meeting 17 : Wed, Sep 18, 02:00 pm-04:00 pm

  Back to Factorization problem. A starting idea to use Fermat's little theorem to extract product of linear factors. Reduction to Squarefree case. <br> Frobenius map (x^q = x) and the sub-algebra of the quotient ring F[x]/f. Dimension of the sub-algebra when f is irreducible.

  References	<ul> <li><a href="http://www.cecm.sfu.ca/CAG/theses/chelsea.pdf">Chelsea's Masters Thesis</a></li> </ul>
  Exercises
  Reading

  Back to Factorization problem. A starting idea to use Fermat's little theorem to extract product of linear factors. Reduction to Squarefree case.
  Frobenius map (x^q = x) and the sub-algebra of the quotient ring F[x]/f. Dimension of the sub-algebra when f is irreducible.
  

  References

  :

  Chelsea's Masters Thesis

  • Meeting 18 : Tue, Sep 24, 02:00 pm-04:00 pm

  Chinese remaindering. Berlekamp algebra W and its dimension. From an element in W to factorization - Berlekamp's Lemma.

  References	<ul> <li><a href="http://www.cecm.sfu.ca/CAG/theses/chelsea.pdf">Chelsea's Masters Thesis</a></li> </ul>
  Exercises
  Reading

  Chinese remaindering. Berlekamp algebra W and its dimension. From an element in W to factorization - Berlekamp's Lemma.
  

  References

  :

  Chelsea's Masters Thesis

  • Meeting 19 : Wed, Sep 25, 02:00 pm-04:00 pm

  Writing a set of linear equations and the Berlekamp Matrix. The O((qn^3)(n^2)(qn^2)) time algorithm. Reducing the first factor of q by faster exponentiation. Removing the second factor of q. Identifying the minimal polynomial for the g(x).

  References	<ul> <li><a href="http://www.cecm.sfu.ca/CAG/theses/chelsea.pdf">Chelsea's Masters Thesis</a></li> </ul>
  Exercises
  Reading

  Writing a set of linear equations and the Berlekamp Matrix. The O((qn^3)(n^2)(qn^2)) time algorithm. Reducing the first factor of q by faster exponentiation. Removing the second factor of q. Identifying the minimal polynomial for the g(x).
  

  References

  :

  Chelsea's Masters Thesis

  • Meeting 20 : Tue, Oct 01, 02:00 pm-04:00 pm

  Computing the minimal polynomial of g(x). Factorizing the minimal polynomial by Rabin's factorization method. Discussions on effect of choosing g(x) in W, at random.

  References	<ul> <li><a href="http://www.cecm.sfu.ca/CAG/theses/chelsea.pdf">Chelsea's Masters Thesis</a></li> </ul>
  Exercises
  Reading

  Computing the minimal polynomial of g(x). Factorizing the minimal polynomial by Rabin's factorization method. Discussions on effect of choosing g(x) in W, at random.
  

  References

  :

  Chelsea's Masters Thesis

  • Meeting 21 : Sat, Oct 05, 10:00 am-01:00 pm

  Distinct Degree Factorization. Basic Idea of the equal degree factorization - finding the zero divisor.

  References	<ul> <li><a href="http://www.cmi.ac.in/~ramprasad/lecturenotes/comp_numb_theory/lecture10.pdf">Lecture 10</a> from <a href="http://www.cse.iitk.ac.in/users/ppk/">Piyush Kurur</a>'s course - notes by <a href="http://www.cmi.ac.in/~ramprasad">Ramprasad Saptarishi</a></li> </ul>
  Exercises
  Reading

  Distinct Degree Factorization. Basic Idea of the equal degree factorization - finding the zero divisor.
  

  References

  :

  Lecture 10 from Piyush Kurur's course - notes by Ramprasad Saptarishi

  • Meeting 22 : Tue, Oct 08, 02:00 pm-03:30 pm

  Cantor-Zassenhaus Algorithm for equal degree factorization. Factorizing over integers and factorizing over rationals. The basis idea using big primes. Hensel Lifting idea.

  References	<ul> <li><a href="http://www.cmi.ac.in/~ramprasad/lecturenotes/comp_numb_theory/lecture11.pdf">Lecture 11</a> from <a href="http://www.cse.iitk.ac.in/users/ppk/">Piyush Kurur</a>'s course - by <a href="http://www.cmi.ac.in/~ramprasad">Ramprasad Saptarishi</a></li> </ul>
  Exercises
  Reading

  Cantor-Zassenhaus Algorithm for equal degree factorization. Factorizing over integers and factorizing over rationals. The basis idea using big primes. Hensel Lifting idea.
  

  References

  :

  Lecture 11 from Piyush Kurur's course - by Ramprasad Saptarishi

  • Meeting 23 : Wed, Oct 09, 02:00 pm-04:00 pm

  Hensel Lifting in detail. Connection to Newton-Raphson method. Factorizing over integers (a trivial algorithm). An Ideal version of Hensel lifting and motivation from the bivariate factorization problem.

  References	<ul> <li>For Hensel Lifting over prime powers : <a href="http://www.cmi.ac.in/~ramprasad/lecturenotes/comp_numb_theory/lecture26.pdf">Lecture 26</a> from <a href="http://www.cse.iitk.ac.in/users/ppk/">Piyush Kurur</a>'s course - by <a href="http://www.cmi.ac.in/~ramprasad">Ramprasad Saptarishi</a></li> <li>For Hensel Lifting over ideals and Applying it in bivariate factorization problem : <a href="http://www.cmi.ac.in/~ramprasad/lecturenotes/algcomp/lecture14.pdf">Lecture 14</a> from <a href="http://www.imsc.res.in/~arvind">V. Arvind</a>'s course - by <a href="http://www.cmi.ac.in/~ramprasad">Ramprasad Saptarishi</a></li> </ul>
  Exercises
  Reading

  Hensel Lifting in detail. Connection to Newton-Raphson method. Factorizing over integers (a trivial algorithm). An Ideal version of Hensel lifting and motivation from the bivariate factorization problem.
  

  References

  :

  For Hensel Lifting over prime powers : Lecture 26 from Piyush Kurur's course - by Ramprasad Saptarishi For Hensel Lifting over ideals and Applying it in bivariate factorization problem : Lecture 14 from V. Arvind's course - by Ramprasad Saptarishi

  • Meeting 24 : Sat, Oct 19, 02:00 pm-04:00 pm

  Factoring bivariate polynomials over finite fields. The algorithm and outline of the correctness argument.

  References	<ul> <li>For Hensel Lifting over ideals and Applying it in bivariate factorization problem : <a href="http://www.cmi.ac.in/~ramprasad/lecturenotes/algcomp/lecture14.pdf">Lecture 14</a> from <a href="http://www.imsc.res.in/~arvind">V. Arvind</a>'s course - by <a href="http://www.cmi.ac.in/~ramprasad">Ramprasad Saptarishi</a></li> </ul>
  Exercises
  Reading

  Factoring bivariate polynomials over finite fields. The algorithm and outline of the correctness argument.
  

  References

  :

  For Hensel Lifting over ideals and Applying it in bivariate factorization problem : Lecture 14 from V. Arvind's course - by Ramprasad Saptarishi

  • Meeting 25 : Tue, Oct 22, 02:00 pm-04:00 pm

  Resultants and their connection to gcds. Correctness proof for the bivariate factorization algorithm.

  References	<ul> <li>For Resultants : Section 1 of <a href="http://www.cmi.ac.in/~ramprasad/lecturenotes/algcomp/lecture13.pdf">Lecture 13</a> from <a href="http://www.imsc.res.in/~arvind">V. Arvind</a>'s course - by <a href="http://www.cmi.ac.in/~ramprasad">Ramprasad Saptarishi</a></li> <li>For the correctness proof of the bivariate factorization : <a href="http://www.cmi.ac.in/~ramprasad/lecturenotes/algcomp/lecture14.pdf">Lecture 14</a> from <a href="http://www.imsc.res.in/~arvind">V. Arvind</a>'s course - by <a href="http://www.cmi.ac.in/~ramprasad">Ramprasad Saptarishi</a></li> </ul>
  Exercises
  Reading

  Resultants and their connection to gcds. Correctness proof for the bivariate factorization algorithm.
  

  References

  :

  For Resultants : Section 1 of Lecture 13 from V. Arvind's course - by Ramprasad Saptarishi For the correctness proof of the bivariate factorization : Lecture 14 from V. Arvind's course - by Ramprasad Saptarishi

  • Meeting 26 : Wed, Oct 23, 02:00 pm-04:00 pm

  Reducing to Squarefree cases in bivariate polynomials. <br> Finding a good substitution for y variable to ensure that the f(x,y) is square free after substituion. <br> Counting the number of irreducible polynomials of a given degree. Back to factoring polynomials over integers.

  References	<ul> <li>For discussion on density of Irreducible Polynomials : see section 1 of <a href="http://www.cmi.ac.in/~ramprasad/lecturenotes/comp_numb_theory/lecture11.pdf">Lecture 11</a> from <a href="http://www.cse.iitk.ac.in/~ppk">Piyush Kurur</a>'s course - by <a href="http://www.cmi.ac.in/~ramprasad">Ramprasad Saptarishi</a></li> <li>For details of the proof of bivariate factorization : <a href="http://www.cmi.ac.in/~ramprasad/lecturenotes/algcomp/lecture15.pdf">Lecture 15</a> from <a href="http://www.imsc.res.in/~arvind">V. Arvind</a>'s course - by <a href="http://www.cmi.ac.in/~ramprasad">Ramprasad Saptarishi</a></li> <li>For Mobius inversion : <a href="http://www.sfu.ca/~mdevos/notes/comb_struct/mobius.pdf">Notes by Matt Devos</a></li> </ul>
  Exercises	Prove the claim from the class: if phi^r(f_1') = f_1', then r must divide t.
  Reading

  Reducing to Squarefree cases in bivariate polynomials.
  Finding a good substitution for y variable to ensure that the f(x,y) is square free after substituion.
  Counting the number of irreducible polynomials of a given degree. Back to factoring polynomials over integers.
  

  References	:	For discussion on density of Irreducible Polynomials : see section 1 of Lecture 11 from Piyush Kurur's course - by Ramprasad Saptarishi For details of the proof of bivariate factorization : Lecture 15 from V. Arvind's course - by Ramprasad Saptarishi For Mobius inversion : Notes by Matt Devos
  Exercises	:	Prove the claim from the class: if phi^r(f_1') = f_1', then r must divide t.

  • Meeting 27 : Sat, Oct 26, 02:00 pm-04:00 pm

  From short vector in lattices. Short Vectors in lattices. The LLL Algorithm (Statement). From short vectors to factoring.

  References
  Exercises
  Reading

  From short vector in lattices. Short Vectors in lattices. The LLL Algorithm (Statement). From short vectors to factoring.
  

  References

  :

  None

  • Meeting 28 : Tue, Oct 29, 02:00 pm-04:00 pm

  Orthogonalization and shortest vector among the orthogonal basis. Reduced Basis. Description of the LLL Algorithm. Main ideas of the proof of running time bound.

  References
  Exercises
  Reading

  Orthogonalization and shortest vector among the orthogonal basis. Reduced Basis. Description of the LLL Algorithm. Main ideas of the proof of running time bound.
  

  References

  :

  None

  • Meeting 29 : Wed, Oct 30, 02:00 pm-04:00 pm

  Detailed proof of run time bound.

  References
  Exercises
  Reading

  Detailed proof of run time bound.
  

  References

  :

  None

• Theme 4 : Algorithms for Identity Testing - 6 meetings

  • Meeting 30 : Tue, Nov 05, 02:00 pm-04:00 pm

  Implicit representation of Polynomials. Arithmetic Circuit Model. Identity Testing Problem.

  References	<ul> <li>Chapter 1 of the survey - <a href="">Arithmetic Circuits - Survey, Recent Results and Open Problems</a> - Shpilka-Yahudayoff</li> </ul>
  Exercises
  Reading

  Implicit representation of Polynomials. Arithmetic Circuit Model. Identity Testing Problem.
  

  References

  :

  Chapter 1 of the survey - Arithmetic Circuits - Survey, Recent Results and Open Problems - Shpilka-Yahudayoff

  • Meeting 31 : Wed, Nov 06, 02:00 pm-04:00 pm

  Identity testing multivariate polynomials. Schwartz-Zippel Lemma. Moshkovitz's proof. Amplification of success probability. Generators vs Hitting Sets.

  References	<ul> <li>Chapter 4 of the survey - <a href="">Arithmetic Circuits - Survey, Recent Results and Open Problems</a> - Shpilka-Yahudayoff</li> </ul>
  Exercises
  Reading

  Identity testing multivariate polynomials. Schwartz-Zippel Lemma. Moshkovitz's proof. Amplification of success probability. Generators vs Hitting Sets.
  

  References

  :

  Chapter 4 of the survey - Arithmetic Circuits - Survey, Recent Results and Open Problems - Shpilka-Yahudayoff

  • Meeting 32 : Thu, Nov 07, 02:00 pm-04:00 pm

  From Hitting Sets to Generators. <br> From Generators to Hitting Sets - Combinatorial Nullstellensatz.<br> Using Algebraic independence to do identity testing. Outline of Chen-Kao algorithm.

  References	<ul> <li>Chapter 4 of the survey - <a href="www.cs.technion.ac.il/~shpilka/publications/SY10.pdf">Arithmetic Circuits - Survey, Recent Results and Open Problems</a> - Shpilka-Yahudayoff</li> </ul>
  Exercises
  Reading

  From Hitting Sets to Generators.
  From Generators to Hitting Sets - Combinatorial Nullstellensatz.
  Using Algebraic independence to do identity testing. Outline of Chen-Kao algorithm.
  

  References

  :

  Chapter 4 of the survey - Arithmetic Circuits - Survey, Recent Results and Open Problems - Shpilka-Yahudayoff

  • Meeting 33 : Sat, Nov 09, 01:30 pm-03:30 pm

  Chen-Kao Algorithm. Error vs Running time Tradeoff.<br> Lewin-Vadhan's algorithm for identity testing over finite fields.

  References	<ul> <li><a href="http://arxiv.org/pdf/cs.DS/9907011">Reducing Randomness via Irrational Numbers</a> - Chen & Kao</li> <li><a href="http://people.seas.harvard.edu/~salil/research/polys-abs.html">Checking Polynomial Identities over any Field: Towards a Derandomization?</a> - Lewin & Vadhan</li> </ul>
  Exercises
  Reading

  Chen-Kao Algorithm. Error vs Running time Tradeoff.
  Lewin-Vadhan's algorithm for identity testing over finite fields.
  

  References

  :

  Reducing Randomness via Irrational Numbers - Chen & Kao Checking Polynomial Identities over any Field: Towards a Derandomization? - Lewin & Vadhan

  • Meeting 34 : Tue, Nov 12, 02:00 pm-04:00 pm

  Identity testing via Chinese remaindering. Some naive attempts and comparison to Schwartz-Zippel Algorithms. Agrawal-Biswas's construction of the sample space using cyclotomic polynomials

  References	<ul> <li><a href="http://repository.ias.ac.in/92018/1/4-p.pdf">Primality and Identity Testing Via Chinese Remaindering</a></li> </ul>
  Exercises
  Reading	<ul> <li><a href="http://www.cse.iitk.ac.in/users/manindra/derandomization/derandomizing-some-identities.pdf">On Derandomizing Tests for Certain Polynomial Identities</a></li> </ul>

  Identity testing via Chinese remaindering. Some naive attempts and comparison to Schwartz-Zippel Algorithms. Agrawal-Biswas's construction of the sample space using cyclotomic polynomials
  

  References	:	Primality and Identity Testing Via Chinese Remaindering
  Reading	:	On Derandomizing Tests for Certain Polynomial Identities

  • Meeting 35 : Wed, Nov 13, 02:00 pm-04:00 pm

  Agrawal-Biswas algorithm. Running and time and error bound. Extensions to multivariate case <br> Randomized algorithm for primality testing - as an identity testing over Rings. Sample space size. <br> Discussion on what we did not cover in the course.

  References	<a href="http://repository.ias.ac.in/92018/1/4-p.pdf">Primality and Identity Testing Via Chinese Remaindering</a> - Agrawal & Biswas
  Exercises
  Reading	<a href="http://www.cse.iitk.ac.in/users/manindra/derandomization/derandomizing-some-identities.pdf">On Derandomizing Tests for Certain Polynomial Identities</a> - Agrawal.

  Agrawal-Biswas algorithm. Running and time and error bound. Extensions to multivariate case
  Randomized algorithm for primality testing - as an identity testing over Rings. Sample space size.
  Discussion on what we did not cover in the course.
  

  References	:	Primality and Identity Testing Via Chinese Remaindering - Agrawal & Biswas
  Reading	:	On Derandomizing Tests for Certain Polynomial Identities - Agrawal.

• Evaluation Meetings - 3 meetings

  • Meeting 36 : Sat, Aug 31, 09:00 am-10:30 am

  Quiz 1 (Lecture 01 - Lecture 10)

  References
  Exercises
  Reading

  Quiz 1 (Lecture 01 - Lecture 10)
  

  References

  :

  None

  • Meeting 37 : Wed, Oct 16, 09:00 am-10:30 am

  Quiz 2 (Lecture 12 - Lecture 23)

  References
  Exercises
  Reading

  Quiz 2 (Lecture 12 - Lecture 23)
  

  References

  :

  None

  • Meeting 38 : Sat, Nov 16, 09:00 am-12:00 pm

  Endsemester Examination

  References
  Exercises
  Reading

  Endsemester Examination
  

  References

  :

  None