I finally found the problem. The form was loaded by ajax.
So, I had to move the script in the file \prestashop\js\admin-products.js
and put it inside the function product_tabs['Informations']
(because my field is under the tab Informations). I also changed the autocomplete script as below.
product_tabs['Informations'] = new function(){
//...
$('#product_type').autocomplete('ajax.php', {
delay: 100,
minChars: 1,
autoFill: true,
max:255,
matchContains: true,
mustMatch:true,
scroll:false,
cacheLength:0,
extraParams: {
ajaxProductType : 1
}
}).result(function(event, item){
$(this).val(item);
});
}