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UID:www.tcs.tifr.res.in/event/903
DTSTAMP:20230914T125942Z
SUMMARY:Constructing Faithful Maps
DESCRIPTION:Speaker: Prerona  Chatterjee\n\nAbstract: \nAbstract: Suppose w
 e are given a set of $k$ polynomials\, $f_1\, \\ldots\, f_k \\in \\mathbb{
 F}[x_1\, \\ldots\, x_n]$. They are said to be algebraically dependent if t
 here is a non-zero polynomial $A$ over $\\mathbb{F}$ in $k$ variables such
  that $A(f_1\, \\ldots\, f_k) = 0$. Otherwise\, they are said to be algebr
 aically independent.\nJust like a set of $k$ linearly independent vectors 
 in $\\mathbb{R}^n$ actually resides in a $k$-dimensional subspace\, a set 
 of $k$ algebraically independent polynomials intuitively has the same "fre
 edom" as only $k$ independent variables. A faithful map formalises this.\
 nSuppose $\\{f_1\, \\ldots\, f_k\\} \\subseteq \\mathbb{F}[x_1\, \\ldots\,
  x_n]$ is a set of algebraically independent polynomials. A map $\\phi : \
 \{x_i\\} \\mapsto \\mathbb{F}[y_1\, \\ldots\, y_k]$ is said to be faithful
  if the set of polynomials $\\{f_i(\\phi(x_1)\, \\ldots\, \\phi(x_n))\\}_{
 i \\in [k]} \\subseteq \\mathbb{F}[y_1\, \\ldots\, y_k]$ is algebraically 
 independent.\nIn this talk\, we will construct faithful maps when the unde
 rlying field is $\\mathbb{Q}$\, $\\mathbb{R}$ or $\\mathbb{C}$ for example
 . Apart from being an interesting question in its own right\, construction
  of faithful maps also has applications in algebraic circuit complexity.\n
URL:https://www.tcs.tifr.res.in/web/events/903
DTSTART;TZID=Asia/Kolkata:20180914T171500
DTEND;TZID=Asia/Kolkata:20180914T181500
LOCATION:A-201 (STCS Seminar Room)
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