Question about "polynomial evaluations"

Brian Chandler wrote:

Stuff addressed to Tim BandTech.com, in which I referred to an example
on these pages from W W Sawyer's little book: http://imaginatorium.org/private/sawyer.gif

I said:
But here you are simply wrong. In particular, in the example Sawyer
gives of polynomials over a finite field ({0, 1} with mod 2
arithmetic, call it F2), the polynomial x^2+x _always_ evaluates to 0.
So in the field of "values" of these polynomial expressions (there
_are_ only two values), these two things are "equal". But in the ring
of polynomials over F2, these are not the same polynomials.
Mathematics is the study of patterns, and is just as free to study the
pattern of *polynomials* as it is to study the pattern of *polynomial
evaluations* (though this is stunningly uninteresting, since it's
exactly the same as the field underlying the polynomials).

The last bit seems to be simply wrong: in this particular case, define
the "evaluation" e(P) of an arbitrary polynomial P as follows.

If we substitute a value for x, this must be 0 or 1, and considering
these two values, there are four cases:

[needs fixed font]

When
x=0 x=1 "Evaluation"
0 0 0
1 1 1
0 1 S (for 'same')
1 0 D (for 'different')

If we define the obvious operations on the Evaluations, e(a) + e(b) = e
(a+b), e(a) * e(b) = e(a*b), we get (I think) that e() is a ring
homomorphism, and these Evaluations must form a ring.

Happily, under addition we get:

+ 0 1 S D
0 0 1 S D
1 1 0 D S
S S D 0 1
D D S 1 0

which looks like the Klein 4-group, but under multiplication we just
seem to get a complicated mess:

* 0 1 S D
0 0 0 0 0
1 0 1 S D
S 0 S S 0
D 0 D 0 D

The questions are: did I make any elementary errors? Is there any
particular significance to the set of "evaluations" as defined?

AIUI, in "ordinary" polynomials, over Z, Q, R [etc?], there is no
meaning to such a set of "evaluations", since two polynomials can only
evaluate to the same value if they are equal as polynomials, and
therefore the "evaluations" simply _are_ the polynomials. Since this
doesn't seem to lead to anything very obvious for the simplest case of
all, perhaps it just explodes for larger fields, and leads nowhere.

Well, any comments would be appreciated.

Brian Chandler
.

Relevant Pages

  • Re: multiplication ofpolynomials using FFT
    ... convolution of the two sequences of coefficients ... At its heart the FFT method replaces the convolution ... calculation by evaluations of the two polynomials at ...
    (sci.math.symbolic)
  • Re: Question about "polynomial evaluations"
    ... of polynomials over F2, these are not the same polynomials. ... polynomial functions as opposed to polynomials. ... If we define the obvious operations on the Evaluations, ... Let R be any ring, and let Rbe the ...
    (sci.math)
  • Re: Understanding the quotient ring nomenclature
    ... On Jun 17, 1:18 am, Brian Chandler ... This is a trivial implication of the fact that a ring is (by ... But if you are going to show some contradiction in mathematics, ... of polynomials over F2, these are not the same polynomials. ...
    (sci.math)
  • Re: Is the polynomial ring interpretation of abstract algebra A[X]
    ... On Jul 2, 11:33 am, Brian Chandler ... polynomial ring actually is. ... this way of looking at polynomials that you seem to have, ... I seek multiple opinions and have received ...
    (sci.math)
  • Polynomial Discriminant
    ... determinant of a matrix which consists of the polynomial coefficients ... the derivative of that polynoimial shifted according to a pattern. ... But I want to have a clue for polynomials of general degree n. ...
    (sci.math)