Chouby on "Polylang : custom post synchrone field problem"

Could you tell me where the "badly synchronized information" is stored?

To answer your other question on debugging, you can put this in your wp-config.php:

define('WP_DEBUG', true);
define('SCRIPT_DEBUG', true);
define('WP_DEBUG_LOG', true);
define('WP_DEBUG_DISPLAY', false);

This will log all errors and warnings in wp-content/debug.log
You can then use error_log function to log strings. You can for example log the content of an array with:

error_log(print_r($my_array, true));