When can you guarantee Minimize outputs the global minimum? [closed]

Question

In the Minimize documentation, it's stated that "Minimize will return exact results if given exact input. With approximate input, it automatically calls NMinimize."

What is an approximate input?

Here is a Mathematica code:

Minimize[-1 - 96 a12^2 - 256 a12^4 - 96 a13^2 - 1536 a12^2 a13^2 - 
  256 a13^4 - 1536 a12 a13 a23 - 96 a23^2 - 1536 a12^2 a23^2 - 
  1536 a13^2 a23^2 - 256 a23^4 + 
  343.04 (1 + 6 a12^2 + a12^4 + 6 a13^2 + 6 a12^2 a13^2 + a13^4 + 
     24 a12 a13 a23 + 6 a23^2 + 6 a12^2 a23^2 + 6 a13^2 a23^2 + 
     a23^4), {a12, a13, a23}]

Output:

{342.04, {a12 -> -5.84843*10^-25, a13 -> -1.68795*10^-23, 
  a23 -> 2.5292*10^-24}}

This is only a local minima. The polynomial attains a lower value, as can be seen by:

-1 - 96 a12^2 - 256 a12^4 - 96 a13^2 - 1536 a12^2 a13^2 - 256 a13^4 - 
    1536 a12 a13 a23 - 96 a23^2 - 1536 a12^2 a23^2 - 
    1536 a13^2 a23^2 - 256 a23^4 + 
    343.04 (1 + 6 a12^2 + a12^4 + 6 a13^2 + 6 a12^2 a13^2 + a13^4 + 
       24 a12 a13 a23 + 6 a23^2 + 6 a12^2 a23^2 + 6 a13^2 a23^2 + 
       a23^4) /. a12 -> -1.9006784725356818 /. 
  a13 -> -1.9006784725356818 /. a23 -> -1.9006784725356818

giving

-520.759

Does Mathematica assume 343.04 to be an approximate input? The following code (where I changed 343.04 to 34304/100 in the above code):

Minimize[-1 - 96 a12^2 - 256 a12^4 - 96 a13^2 - 1536 a12^2 a13^2 - 
  256 a13^4 - 1536 a12 a13 a23 - 96 a23^2 - 1536 a12^2 a23^2 - 
  1536 a13^2 a23^2 - 256 a23^4 + 
  34304/100 (1 + 6 a12^2 + a12^4 + 6 a13^2 + 6 a12^2 a13^2 + a13^4 + 
     24 a12 a13 a23 + 6 a23^2 + 6 a12^2 a23^2 + 6 a13^2 a23^2 + 
     a23^4), {a12, a13, a23}]

seems to give the global minimum :

{-1 + (2 (9666857885 - 82223496 Sqrt[62862]))/
  42128975, {a12 -> 
   Root[{-786971886042052 - 966685788500 # + 1053224375 #^2& , 
     2204297 + 425 # + 8020656 #2 + 7260432 #2^2 + 2134656 #\
2^3 + 129472 #2^4& }, {1, 3}], 
  a13 -> Root[{-786971886042052 - 966685788500 # + 1053224375 #^2& , 
     2204297 + 425 # + 8020656 #2 + 7260432 #2^2 + 2134656 #\
2^3 + 129472 #2^4& , \
-1018313735738200449659774784 + 5698645395376234941626400 #\
 - 11083614190038055995575404032 #2^2 + 1986326192315901090412800 # #\
2^2 - 8185110530032055693037662208 #2^4 + 611388495811547593574400 # #\
2^4 - 3048372220893668754063261696 #2^6 + 90939810999678739660800 # #\
2^6 - 600354216916620229531926528 #2^8 + 16135439394594009907200 # #\
2^8 - 63323209195296300000608256 #2^10 - 1872571526540616829763584 #\
2^12 - 11083614190038055995575404032 #3^2 + 1986326192315901090412800 #\
 #3^2 + 4287462015822562049825673216 #2^2 #\
3^2 + 25764211964617179859814400 # #2^2 #\
3^2 - 36816286729444655930532790272 #2^4 #\
3^2 + 6807253648032933916262400 # #2^4 #\
3^2 - 11381778914420444732185116672 #2^6 #\
3^2 + 24203159091891014860800 # #2^6 #\
3^2 - 348277650574129650003345408 #2^8 #\
3^2 - 16853143738865551467872256 #2^10 #\
3^2 - 8185110530032055693037662208 #3^4 + 611388495811547593574400 # #\
3^4 - 36816286729444655930532790272 #2^2 #\
3^4 + 6807253648032933916262400 # #2^2 #\
3^4 + 77912326749562233967540174848 #2^4 #\
3^4 + 32270878789188019814400 # #2^4 #\
3^4 - 3034907126649757422292303872 #2^6 #\
3^4 - 43537287992069341292003328 #2^8 #\
3^4 - 3048372220893668754063261696 #3^6 + 90939810999678739660800 # #\
3^6 - 11381778914420444732185116672 #2^2 #\
3^6 + 24203159091891014860800 # #2^2 #\
3^6 - 3034907126649757422292303872 #2^4 #\
3^6 - 58986003086029430137552896 #2^6 #\
3^6 - 600354216916620229531926528 #3^8 + 16135439394594009907200 # #\
3^8 - 348277650574129650003345408 #2^2 #\
3^8 - 43537287992069341292003328 #2^4 #\
3^8 - 63323209195296300000608256 #3^10 - 16853143738865551467872256 #\
2^2 #3^10 - 1872571526540616829763584 #3^12& }, {1, 3, 2}], 
  a23 -> Root[{-786971886042052 - 966685788500 # + 1053224375 #^2& , 
     2204297 + 425 # + 8020656 #2 + 7260432 #2^2 + 2134656 #\
2^3 + 129472 #2^4& , \
-1018313735738200449659774784 + 5698645395376234941626400 #\
 - 11083614190038055995575404032 #2^2 + 1986326192315901090412800 # #\
2^2 - 8185110530032055693037662208 #2^4 + 611388495811547593574400 # #\
2^4 - 3048372220893668754063261696 #2^6 + 90939810999678739660800 # #\
2^6 - 600354216916620229531926528 #2^8 + 16135439394594009907200 # #\
2^8 - 63323209195296300000608256 #2^10 - 1872571526540616829763584 #\
2^12 - 11083614190038055995575404032 #3^2 + 1986326192315901090412800 #\
 #3^2 + 4287462015822562049825673216 #2^2 #\
3^2 + 25764211964617179859814400 # #2^2 #\
3^2 - 36816286729444655930532790272 #2^4 #\
3^2 + 6807253648032933916262400 # #2^4 #\
3^2 - 11381778914420444732185116672 #2^6 #\
3^2 + 24203159091891014860800 # #2^6 #\
3^2 - 348277650574129650003345408 #2^8 #\
3^2 - 16853143738865551467872256 #2^10 #\
3^2 - 8185110530032055693037662208 #3^4 + 611388495811547593574400 # #\
3^4 - 36816286729444655930532790272 #2^2 #\
3^4 + 6807253648032933916262400 # #2^2 #\
3^4 + 77912326749562233967540174848 #2^4 #\
3^4 + 32270878789188019814400 # #2^4 #\
3^4 - 3034907126649757422292303872 #2^6 #\
3^4 - 43537287992069341292003328 #2^8 #\
3^4 - 3048372220893668754063261696 #3^6 + 90939810999678739660800 # #\
3^6 - 11381778914420444732185116672 #2^2 #\
3^6 + 24203159091891014860800 # #2^2 #\
3^6 - 3034907126649757422292303872 #2^4 #\
3^6 - 58986003086029430137552896 #2^6 #\
3^6 - 600354216916620229531926528 #3^8 + 16135439394594009907200 # #\
3^8 - 348277650574129650003345408 #2^2 #\
3^8 - 43537287992069341292003328 #2^4 #\
3^8 - 63323209195296300000608256 #3^10 - 16853143738865551467872256 #\
2^2 #3^10 - 1872571526540616829763584 #3^12& , \
-8576 + 25 # - 49056 #2^2 - 2176 #2^4 - 49056 #3^2 - 13056 #2^2 #\
3^2 - 2176 #3^4 - 167424 #2 #3 #4 - 49056 #4^2 - 13056 #2^2 #\
4^2 - 13056 #3^2 #4^2 - 2176 #4^4& }, {1, 3, 2, 2}]}}

Is this indeed the global minimum?