From: Dave Rusin <rusin@math.niu.edu>
Subject: Re: Hit Polynomials
Date: Sat, 15 May 1999 02:57:28 -0500 (CDT)
Newsgroups: sci.math.research
To: tan_khengfeung@pacific.net.sg
Keywords: Action of Steenrod algebra on polynomial rings

In article <37390B45.633D57FD@pacific.net.sg> you write:
>Let GL_2 denote the set of all invertible matrices over F_2, the field
>with 2 elements. This set acts on the polynomial ring
>P_3=F_2[x_1,x_2,x_3]. The set of invariants under this action is again a
>polynomial ring generated by Q_{3,0}, Q_{3,1} and Q_{3,2}. Let D_3
>denote this polynomial ring. Since the Steenrod squares act on P_3, and
>their action commutes with those of GL_2, we can consider the action of
>the Steenrod squares on D_3.

OK - Dickson invariants. I guess you've read Wilkerson's "Primer"?

>We say that a polynomial, F, in P_3 is hit in P_3 if we can write F as a
>linear combination of Sq^i(f_i), for appropriate f_i in P_3.

I'm assuming you mean, "...for i > 0", since  Sq^0  is the identity map.

>I have come across many results concerning hit polynomials. Here are
>some questions:
>
>1) Are all the elements in D_3, regarded as a subset of P_3, hit in P_3?
>If so, where can I get a proof of this result?

Maybe. I'll have to think about this. Let me know if you get any replies.

>2) I have attempted  to show that the element Q_{3,0}Q_{3,1}Q_{3,2} is
>hit in P_3. However, in order to do so, I have to re-express it in terms
>of x_1, x_2 and x_3. This is a very tedious exercise (I'm working by
>hand!). Hence I would like to know if there are any easier way to show
>that this element is hit (I am quite convinced that this element is
>hit).

It is "hit". In a completely uninspired calculation, I had Maple look
for something which would hit it. It's Sq^1(f1) + ... + Sq^7(f7)  where
I write the  f_i  in terms of  X=x_1, Y=x_2, and Z=x_3  for simplicity:

X^9*Y^6*Z+X^3*Y^12*Z:

X^3*Y^9*Z^3+X^9*Y^3*Z^3+X^3*Y^3*Z^9:

X*Y^7*Z^6+X^2*Y^9*Z^3+X^4*Y^3*Z^7+X^9*Y^3*Z^2+X^7*Y^6*Z+X^7*Y^5*Z^2+
     X*Y^6*Z^7+X^2*Y^7*Z^5:

X^5*Y^6*Z^2+X^2*Y^5*Z^6+X^4*Y^6*Z^3:

X^5*Y^2*Z^5+X^3*Y^6*Z^3+X^2*Y^5*Z^5+X*Y^4*Z^7+X^4*Y^5*Z^3+X^6*Y^3*Z^3+
 X^3*Y^3*Z^6+X^7*Y^3*Z^2+X*Y^6*Z^5+X^2*Y^3*Z^7+X^2*Y^7*Z^3+X*Y^7*Z^4+X^7*Y^4*Z:

X^4*Y*Z^6+X^3*Y^6*Z^2+X^6*Y*Z^4+X^5*Y*Z^5+X*Y^5*Z^5+X^2*Y^6*Z^3+
    X^6*Y^3*Z^2+X^2*Y^3*Z^6+X^5*Y^5*Z:

X^5*Y^2*Z^3+X^3*Y^2*Z^5:

>3) Can anybody tell me what is the latest progress on the problem of
>finding hit polynomials?

Is this a known problem? How does it arise?
dave

WW := [[X^9*Y^6*Z+X^3*Y^12*Z, 1], [X^3*Y^9*Z^3+X^9*Y^3*Z^3+X^3*Y^3*Z^9, 2], [X
*Y^7*Z^6+X^2*Y^9*Z^3+X^4*Y^3*Z^7+X^9*Y^3*Z^2+X^7*Y^6*Z+X^7*Y^5*Z^2+X*Y^6*Z^7+X
^2*Y^7*Z^5, 3], [X^5*Y^6*Z^2+X^2*Y^5*Z^6+X^4*Y^6*Z^3, 4], [X^5*Y^2*Z^5+X^3*Y^6
*Z^3+X^2*Y^5*Z^5+X*Y^4*Z^7+X^4*Y^5*Z^3+X^6*Y^3*Z^3+X^3*Y^3*Z^6+X^7*Y^3*Z^2+X*Y
^6*Z^5+X^2*Y^3*Z^7+X^2*Y^7*Z^3+X*Y^7*Z^4+X^7*Y^4*Z, 5], [X^4*Y*Z^6+X^3*Y^6*Z^2
+X^6*Y*Z^4+X^5*Y*Z^5+X*Y^5*Z^5+X^2*Y^6*Z^3+X^6*Y^3*Z^2+X^2*Y^3*Z^6+X^5*Y^5*Z,
6], [X^5*Y^2*Z^3+X^3*Y^2*Z^5, 7]];