Multi-branched polyalkylene glycol polymer, method for producing the same, and admixture for cement

A multi-branched polymer, comprising:

• a polyalkylene glycol chain;
• a residue of a compound having three or more active hydrogen atoms;
• and
• a polymer portion having a constitutional unit derived from a vinyl monomer component,
• one end of the polyalkylene glycol chain bonding to the residue, and
• a terminal oxygen atom of at least one of the other ends of the polyalkylene glycol chain bonding to a main chain end of the polymer portion directly or via an organic residue, wherein said multi-branched polymer is represented by the following formula (1):
• wherein X is a residue of the compound having three or more active hydrogen atoms;
• each AO is a C2-C18 oxyalkylene group, and the AOs may be the same as or different from each other;
• each Y1 is a direct bond or an organic residue;
• each Z is a polymer portion having a constitutional unit derived from a vinyl monomer component;
• each n is the average number of moles of the oxyalkylene groups added and is 5 to 1,000, and the ns may be the same as or different from each other;
• and m is an integer of 3 to 50.