From: Dave Rusin 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]];