a potential undecidable problem related to a Hodge theoretic generalization of Jacobi's formula
José Manuel Rodríguez Caballero
josephcmac at gmail.com
Sun Sep 11 15:48:12 EDT 2022
Dear FOM members,
In the framework of Stephen Wolfram's physicalization of metamathematics,
https://arxiv.org/abs/2204.05123
in the same way that the classical physical world that we observe in
everyday life arises from a quantum physical world, there should be a
quantum version of mathematics from which classical mathematics can be
obtained as a limit (just like classical mechanics is the limit of quantum
mechanics). Of course, many versions of quantum math can be created
trivially using q-series and q-analog. The real problem is to find an
interesting version of quantum math, in which the important theorems from
classical math can be generalized in an insightful way.
One of the tips to develop interesting quantum mathematics is to obtain the
number theory as a limit of the finite field theory when the number of
elements of the field approaches 1. Of course, such a statement is absurd
in the general framework of finite fields, since the number of elements can
only be a prime power. However, this logical impossibility did not prevent
Jacques Tits from developing the F1 heuristic. Alain Connes is pursuing
this approach (F1 heuristics) to try to express the Riemann hypothesis
(part of classical mathematics) as the limit of the Weil conjectures (part
of quantum mathematics):
Connes, Alain, Caterina Consani, and Matilde Marcolli. "Fun with F1." *Journal
of Number Theory* 129.6 (2009): 1532-1561.
https://www.sciencedirect.com/science/article/pii/S0022314X08001856
In the framework of Weil's conjectures, applying Lefschetz's fixed-point
theorem to the Frobenius endomorphism we see that quantum mathematics,
developed from the F1 heuristic, is deeply linked to cohomology. Studying
the mixed Hodge structure of the Hibert scheme of n points on an algebraic
torus, Hausel, Letellier, and Rodriguez-Villegas
Hausel, Tamás, Emmanuel Letellier, and Fernando Rodriguez-Villegas.
"Arithmetic harmonic analysis on character and quiver varieties II." *Advances
in Mathematics* 234 (2013): 85-128.
https://www.sciencedirect.com/science/article/pii/S0001870812004008
basically obtained in (4.1.3) a mixed Hodge polynomial that I think is
connected to some undecidable problems in a deep way. Doing algebraic
manipulations to this polynomial, I obtained a generalization of Jacobi's
formula for the number of ways the number n can be expressed as the sum of
two squares (A004018 in OEIS, denoted r2(n))
https://arxiv.org/pdf/1801.03134.pdf
My q-analog of r2(n) (for simplicity I will use the same notation, but it
is a polynomial, not a number) can be computed in the R programming
language by means of the following code corresponding to formulae (5), (6)
and (7) in my preprint:
library(numbers)
library(polynom)
library(purrr)
x <- polynomial(coef = c(0, 1))
f <- function(m, d) {
return ( x^( (m/d+1)*(d-1)/2 ) + x^( (m/d-1)*(d+1)/2 ) )
}
r2 <- function(n) {
dAll <- divisors(n)
m <- 2*n
d1 <- dAll[dAll %% 4 == 1]
p1 <- (x+1)*Reduce(`+`, map(d1, function(d) f(m, d) ))
remove(d1)
d3 <- dAll[dAll %% 4 == 3]
remove(dAll)
if (length(d3) == 0) {
return(p1)
} else {
p3 <- (x+1)*Reduce(`+`, map(d3, function(d) f(m, d)))
remove(d3)
return( p1 - p3 )
}
}
For example, r2(15) is
1 + x - x^8 - x^9 - x^11 - x^12 + x^14 + 2*x^15 + x^16 - x^18 - x^19 - x^21
- x^22 + x^29 + x^30
Notice that the set of non-zero coefficients of r2(15) is
1, 1, -1, -1, -1, -1, 1, 2, 1, -1, -1, -1, -1, 1, 1
Now, the problem that I think it is uncomputable is the following: given a
finite sequence of integers (like the sequence above), decide if there is a
positive integer n for which these numbers are the list of the non-zero
coefficients of the polynomial r2(n). The only reference that I found that
may give some clue on how to think about this problem is the work of Gerald
Tenenbaum and co-authors concerning the distribution of divisors of a
"typical" integer, for example,
Hall, Richard Roxby, and Gérald Tenenbaum. *Divisors*. Cambridge University
Press, 1988.
https://www.amazon.ca/Divisors-Richard-R-Hall/dp/0521091675
Nevertheless, non-typical integers (in the sense of probabilistic number
theory) may be crucial for solving the decidability problem described
above. Any suggestion on how to encode the halting problem in the framework
of the divisors of an integer that may be relevant to solve this problem?
Kind regards,
Jose M.
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