Comments on: Distinguished Lecture Series III: Avi Wigderson, “Algebraic computation” http://terrytao.wordpress.com/2008/01/13/distinguished-lecture-series-iii-avi-wigderson-algebraic-computation/ Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao Thu, 07 Aug 2008 21:46:04 +0000 http://wordpress.org/?v=MU By: Terence Tao http://terrytao.wordpress.com/2008/01/13/distinguished-lecture-series-iii-avi-wigderson-algebraic-computation/#comment-24562 Terence Tao Wed, 16 Jan 2008 20:35:27 +0000 http://terrytao.wordpress.com/2008/01/13/distinguished-lecture-series-iii-avi-wigderson-algebraic-computation/#comment-24562 Thanks for the reference and the correction! It's easy enough to fix, of course, by shifting all the $latex x_j^d$ by 1, which I've now done. Thanks for the reference and the correction! It’s easy enough to fix, of course, by shifting all the x_j^d by 1, which I’ve now done.

]]> By: Eric http://terrytao.wordpress.com/2008/01/13/distinguished-lecture-series-iii-avi-wigderson-algebraic-computation/#comment-24547 Eric Wed, 16 Jan 2008 17:58:55 +0000 http://terrytao.wordpress.com/2008/01/13/distinguished-lecture-series-iii-avi-wigderson-algebraic-computation/#comment-24547 "Algebraic complexity theory", by Buergisser, Clausen, Shokrollahi is a good reference. It contains most of the algebra needed for the degree bound (by the way, the variety {x1^d=...=xn^d=0} has degree 1, since it's a point) “Algebraic complexity theory”, by Buergisser, Clausen, Shokrollahi is a good reference. It contains most of the algebra needed for the degree bound (by the way, the variety {x1^d=…=xn^d=0} has degree 1, since it’s a point)

]]> By: Terence Tao http://terrytao.wordpress.com/2008/01/13/distinguished-lecture-series-iii-avi-wigderson-algebraic-computation/#comment-24480 Terence Tao Wed, 16 Jan 2008 02:44:55 +0000 http://terrytao.wordpress.com/2008/01/13/distinguished-lecture-series-iii-avi-wigderson-algebraic-computation/#comment-24480 Dear Jason, Using the recurrence described above, one can get from $latex \hbox{Sym}_{n-1}^0, \ldots, \hbox{Sym}_{n-1}^d$ to $latex \hbox{Sym}_n^0, \ldots, \hbox{Sym}_n^d$ using a circuit of complexity O(d). Iterating this n times gives the claim. I learned algebraic geometry from a number of texts, from which I found Harris's text to be the most accessible. But perhaps others will have further suggestions. Dear Jason,

Using the recurrence described above, one can get from \hbox{Sym}_{n-1}^0, \ldots, \hbox{Sym}_{n-1}^d to \hbox{Sym}_n^0, \ldots, \hbox{Sym}_n^d using a circuit of complexity O(d). Iterating this n times gives the claim.

I learned algebraic geometry from a number of texts, from which I found Harris’s text to be the most accessible. But perhaps others will have further suggestions.

]]> By: Jason http://terrytao.wordpress.com/2008/01/13/distinguished-lecture-series-iii-avi-wigderson-algebraic-computation/#comment-24476 Jason Wed, 16 Jan 2008 02:30:41 +0000 http://terrytao.wordpress.com/2008/01/13/distinguished-lecture-series-iii-avi-wigderson-algebraic-computation/#comment-24476 Hi Terry, Can you show in more detail why $latex S(Sym_n^d) = O(nd)$? I am still not clear from your hints that "By recursively describing the symmetric polynomials of degree up to d in terms of the same polynomials on one fewer variable, one can obtain an upper bound $latex S(Sym_n^d) = O(nd)$" As I see it, $latex Sym_n^d = Sym_{n-1}^d + x_n Sym_{n-1}^{d-1}$. but can this recurrence result in an $latex O(nd)$ size circuit? Moreover, since I'm computer science major, the notion "algebraic geometry" terrifies me. Can you provide any good textbooks about it, in particular suitbable for non-math major students? Hi Terry,

Can you show in more detail why S(Sym_n^d) = O(nd)? I am still not clear from your hints that

“By recursively describing the symmetric polynomials of degree up to d in terms of the same polynomials on one fewer variable, one can obtain an upper bound S(Sym_n^d) = O(nd)”

As I see it,

Sym_n^d = Sym_{n-1}^d + x_n Sym_{n-1}^{d-1}.

but can this recurrence result in an O(nd) size circuit?

Moreover, since I’m computer science major, the notion “algebraic
geometry” terrifies me. Can you provide any good textbooks about it,
in particular suitbable for non-math major students?

]]> By: Iftikhar Burhanuddin http://terrytao.wordpress.com/2008/01/13/distinguished-lecture-series-iii-avi-wigderson-algebraic-computation/#comment-24348 Iftikhar Burhanuddin Tue, 15 Jan 2008 02:42:36 +0000 http://terrytao.wordpress.com/2008/01/13/distinguished-lecture-series-iii-avi-wigderson-algebraic-computation/#comment-24348 I've made available the audio recordings and pictures of Avi Wigderson's DLS series of talks here: http://www.math.ucla.edu/~ntg/#seminar I’ve made available the audio recordings and pictures of Avi Wigderson’s DLS series of talks here:

http://www.math.ucla.edu/~ntg/#seminar

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