![](/files/happy5.png)
![$ B_1,B_2,\ldots B_p $](/files/tex/d7626d3626b2054ebc198940785a7861d2fae9c2.png)
![$ n \times n $](/files/tex/fd981d449b91b1f4889d87406e6aa7d8acfb5d68.png)
![$ {\mathbb Z}_p $](/files/tex/e8c94ceb5a9d688bff114c12f7fe9fe47ef955fc.png)
![$ p $](/files/tex/928cd9d544fdea62f88a627aaee28c416c4366c0.png)
![$ n \times (p-1)n $](/files/tex/18102393d42ad781eb0253bf9bee94b60757ed23.png)
![$ A $](/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png)
![$ [B_1 B_2 \ldots B_p] $](/files/tex/86661dc2948aeca789b4392c2e2a9cbf7d96f735.png)
![$ A $](/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png)
Definition: If is an
matrix over a field of characteristic
, then we say that
is an Alon-Tarsi basis (or AT-basis) if the permanent of the
matrix obtained by stacking
copies of
is nonzero.
It follows from the Alon-Tarsi polynomial technique that if is an AT-base then for every
of size 2 and for every
, there exists a vector
so that
(using the notation from A nowhere-zero point in a linear mapping,
is (2,1)-choosable). It follows from this that every Alon-Tarsi base over
is also an additive basis. Thus, the above conjecture, if true, would imply The additive basis conjecture. The following strengthening of this conjecture was suggested in [D]
![$ B_1,B_2,\ldots,B_p $](/files/tex/1f7076819a8c6ddb20523292fafd6bf923742665.png)
![$ n \times n $](/files/tex/fd981d449b91b1f4889d87406e6aa7d8acfb5d68.png)
![$ p $](/files/tex/928cd9d544fdea62f88a627aaee28c416c4366c0.png)
![$ [B_1 B_2 \ldots B_p] $](/files/tex/86661dc2948aeca789b4392c2e2a9cbf7d96f735.png)
![$ n \times (p-1)n $](/files/tex/18102393d42ad781eb0253bf9bee94b60757ed23.png)
![$ A $](/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png)
![$ n \times n $](/files/tex/fd981d449b91b1f4889d87406e6aa7d8acfb5d68.png)
![$ C $](/files/tex/05d3558ef52267cc1af40e658352d98883668ee3.png)
![$ A $](/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png)
![$ C $](/files/tex/05d3558ef52267cc1af40e658352d98883668ee3.png)
In addition to implying the conjecture, above, if true, this conjecture would imply both The permanent conjecture and The choosability in conjecture.