symbolic modular integers still broken

[sagetrac-tmonteil]

sage: f(x) = Zmod(7)(1) * x**2 + Zmod(9)(1) * x**3
sage: f(1)
2
sage: Zmod(7)(1) + Zmod(9)(1)

...
TypeError: unsupported operand parent(s) for '+': 'Ring of integers modulo 7' and 'Ring of integers modulo 9'

}}}

Depends on #21391

CC: @kcrisman @slel

Component: symbolics

Branch/Commit: u/rws/symbolic_modular_integers_still_broken @ be9ec4e

Reviewer: Ralf Stephan

Issue created by migration from https://trac.sagemath.org/ticket/18787

[jdemeyer]

Description changed:

--- 
+++ 
@@ -1,9 +1,8 @@
 As reported on [this ask question](http://ask.sagemath.org/question/27196/multiplying-matrices-with-different-parents/)
 
 ```
-sage: sage: var('A,B,C,D')
-(A, B, C, D)
-sage: S = matrix(SR,2,[[1+3*A,3*B],[3*C,1+3*D]])
+sage: _ =var('A,B,C,D')
+sage: S = matrix([[1+3*A,3*B],[3*C,1+3*D]])
 sage: R = Zmod(9)
 sage: g = matrix(R,2,[[2,1],[2,6]])
 sage: gi = matrix(R,2,[[6,8],[7,2]])
@@ -15,12 +14,12 @@
 While we have (look at the second line):
 
 ```
-sage: R.<A,B,C,D> = PolynomialRing(Zmod(9),4)
+sage: P.<A,B,C,D> = ZZ[]
+sage: S = matrix([[1+3*A,3*B],[3*C,1+3*D]])
+sage: R = Zmod(9)
 sage: g = matrix(R,2,[[2,1],[2,6]])
 sage: gi = matrix(R,2,[[6,8],[7,2]])
-sage: S = matrix(R,2,[[1+3*A,3*B],[3*C,1+3*D]])
 sage: g*S*gi
 [        6*B + 3*D + 1 3*A + 3*B + 6*C + 6*D]
 [                  6*B         3*A + 3*B + 1]
 ```
-

[jdemeyer]

Description changed:

--- 
+++ 
@@ -1,25 +1,7 @@
 As reported on [this ask question](http://ask.sagemath.org/question/27196/multiplying-matrices-with-different-parents/)
 
 ```
-sage: _ =var('A,B,C,D')
-sage: S = matrix([[1+3*A,3*B],[3*C,1+3*D]])
-sage: R = Zmod(9)
-sage: g = matrix(R,2,[[2,1],[2,6]])
-sage: gi = matrix(R,2,[[6,8],[7,2]])
-sage: g*S*gi
-[        6*B + 3*D + 1 3*A + 3*B + 6*C + 6*D]
-[            0*A + 0*D         3*A + 0*D + 1]
+sage: _ = var('A,B,C,D')
+sage: (B + 3*D)*Zmod(9)(6)
+0*D
 ```
-
-While we have (look at the second line):
-
-```
-sage: P.<A,B,C,D> = ZZ[]
-sage: S = matrix([[1+3*A,3*B],[3*C,1+3*D]])
-sage: R = Zmod(9)
-sage: g = matrix(R,2,[[2,1],[2,6]])
-sage: gi = matrix(R,2,[[6,8],[7,2]])
-sage: g*S*gi
-[        6*B + 3*D + 1 3*A + 3*B + 6*C + 6*D]
-[                  6*B         3*A + 3*B + 1]
-```

[sagetrac-tmonteil]

comment:4

Thanks for isolating the problem.

[sagetrac-tmonteil]

Description changed:

--- 
+++ 
@@ -1,7 +1,10 @@
 As reported on [this ask question](http://ask.sagemath.org/question/27196/multiplying-matrices-with-different-parents/)
 
 ```
-sage: _ = var('A,B,C,D')
-sage: (B + 3*D)*Zmod(9)(6)
-0*D
+sage: _ = var('A,B')
+sage: (A + 3*B)*Zmod(9)(6)
+0*B
 ```
+
+while the result should be `6*A`.
+

[kcrisman]

comment:5

Thanks very much to both for reporting and simplifying this.

[nbruin]

comment:6

A different issue, but it might be related:

sage: F=sum((i+1)*x^i for i in [0..20])
sage: G=sum(Zmod(7)(i+1)*x^i for i in [0..20])
sage: F*Zmod(7)(1) - G
x^19 + 2*x^18 + 0*x^13 + 0*x^6
sage: G*Zmod(7)(1) - G
0

so there seems to be something fishy with symbolic multiplication involving integers and elements of Z/n in general. It's not just a zero-divisor problem.

It's not just powers either:

sage: V=[SR.var("x%s"%i) for i in [0..20]]
sage: F=sum((i+1)*V[i] for i in [0..20])
sage: G=sum(Zmod(7)(i+1)*V[i] for i in [0..20])
sage: sage: F*Zmod(7)(1)-G
-x0 + 0*x13 + 6*x14 + 0*x20 + 3*x3 + 0*x6
sage: F-G 
14*x13 + 21*x20 + 7*x6

The last one is correct, since the terms with x6, x20 and x13 are really missing from G. This illustrates why mixing characteristics in SR is always going to be a mess (even if non-zero characteristic works properly otherwise).

[sagetrac-tmonteil]

comment:7

It seems to be a pynac/ginac issue, right ? So what should we do on our side ? Add a stopgap during conversion/coercion Zmod(n) -> SR ?

[kcrisman]

comment:8

(Or even just disallow it?)

[rwst]

comment:9

This is working in pynac-0.4.1 (but not 0.3.9.1).

EDIT: I meant the original case.

[rwst]

comment:10

Replying to @nbruin:

It's not just powers either:

sage: V=[SR.var("x%s"%i) for i in [0..20]]
sage: F=sum((i+1)*V[i] for i in [0..20])
sage: G=sum(Zmod(7)(i+1)*V[i] for i in [0..20])
sage: sage: F*Zmod(7)(1)-G
-x0 + 0*x13 + 6*x14 + 0*x20 + 3*x3 + 0*x6
sage: F-G 
14*x13 + 21*x20 + 7*x6

This is different in 0.4.1:

sage: sage: sage: F*Zmod(7)(1)-G
-x0 + 0*x13 + 0*x20 + 4*x9
sage: F-G
14*x13 + 21*x20 + 7*x6
sage: F
x0 + 2*x1 + 11*x10 + 12*x11 + 13*x12 + 14*x13 + 15*x14 + 16*x15 + 17*x16 + 18*x17 + 19*x18 + 20*x19 + 3*x2 + 21*x20 + 4*x3 + 5*x4 + 6*x5 + 7*x6 + 8*x7 + 9*x8 + 10*x9
sage: G
x0 + 2*x1 + 4*x10 + 5*x11 + 6*x12 + x14 + 2*x15 + 3*x16 + 4*x17 + 5*x18 + 6*x19 + 3*x2 + 4*x3 + 5*x4 + 6*x5 + x7 + 2*x8 + 3*x9

EDIT: Added F,G.