To make a long story short; you risk to either be drowning, get eaten or have your neck broken.
The best protection against polar bears is a gun. Shotgun is the standard of Canada and a Magnum handheld the choice of Russia. The shotgun should be 15 calibers and not greased. Keep it outside the tent at all times or it will ice up. Tie a string to it leading into the tent for fast reach. Protect it in a durable plastic bag and carry it in a padded gun cover with a shoulder strap. Carry it with you at all times, don't leave it on one sled when returning for the other. Mantle and test fire occasionally. The gun provided at Resolute Bay with names carved in its handle has a history of malfunction, especially in very cold temperatures. We advise to stay away from it.
There are flares available to scare off the bear. The accounts for their use have however been somewhat disappointing. The bears have ducked and then kept on with their business of mugging sleds.
Rubber bullets are the best bear scare offs, yet if the bear still refuses to budge you'll have to shoot it. Remember that it is a criminal offence to shoot a bear without proper procedure (trying to scare it off at first) and especially if it is shot at long distance in its back as in the case of one polar expedition.
Should your gun refuse to fire, your flares not do the trick and your barking like a dog just enhance the bears interest in you, there is the last resort of bear spray. The weak part of the bear spray can is the aerosol pump that can freeze in cold temperatures. Keep the spray inside your clothing while traveling and inside your sleeping bag at night. We carried it on us, then pulled it out and tested it in below -40C. It worked fine every time.
Try the spray can out beforehand; in the event of a polar bear attack there is no time to read instructions. Spray away from the wind and keep your cool, the bear must be just 6 feet away for the spray to have effect. Polar bears are not likely to charge at you, they will rather just trot towards you and that will give you enough time to react. Don't panic. There is at least one polar trekker (Japanese) that has sprayed a bear to flight. There are to this day no known Polar bear fatalities involving North Pole trekkers.
Store a spray can close at hand inside your sleeping bag at night. Sweet dreams..:)