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On my ballot for Minnesota are a lot of state-level positions (judges, a representative, county sheriff) where the incumbent is running unopposed. If there are fewer votes for them than blank votes, would they be elected?

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Yes. An unopposed candidate is elected if that candidate receives at least one vote.

A blank vote, also known as an "undervote" does not count and is equivalent to not casting a ballot at all with respect to that office. There is no quorum of minimum support. An undervote is not treated as a vote against an unopposed candidate in Minnesota.

The only U.S. state in which there is an actual "None of these Candidates" option on the ballot, rather than the mere reality of undervotes, is Nevada, which has had this option since 1976. And, even there, a candidate will not lose if there are more "None of these Candidates" receives more vote than the candidate receiving the highest number of votes. As Wikipedia explains:

In the event that the "None of These Candidates" option receives the most votes in an election, the actual candidate who receives the most votes still wins the election. This has most notably happened on two occasions: in the 1976 Republican primary for Nevada's At-large congressional district, None of These Candidates received 16,097 votes, while Walden Earhart won 9,831 votes, followed by Dart Anthony with 8,097 votes. Even though he received fewer votes than "None of These Candidates", Earhart received the Republican nomination. He went on to lose to incumbent Democratic Congressman Jim Santini in the general election. In the 2014 Democratic gubernatorial primary, "None of These Candidates" won 30% of the popular vote, a plurality. Robert Goodman, the runner-up with 25% of the vote, was the Democratic nominee by state law.

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Yes. Minnesota election laws for state-wide offices seem to be at section 208.05 STATE CANVASSING BOARD. That statute isn't a model of clarity, but parsing it logically it seems what matters is the number of votes received, not undervotes.

The state canvassing board at its meeting on the date provided in section 204C.33 shall open and canvass the returns made to the secretary of state for presidential electors and alternates, prepare a statement of the number of votes cast for the persons receiving votes for these offices, and declare the person or persons receiving the highest number of votes for each office duly elected.

P.S. -- The above passage may in be in scope only for presidential elections, in which case section 204C.33 CANVASS OF STATE GENERAL ELECTIONS control for the examples given. This logically equivalent portion of the statutes probably control:

Upon completion of the canvass, the county canvassing board shall declare the candidate duly elected who received the highest number of votes for each county and state office voted for only within the county.

It appears the only substantial difference is at what level of government the process occurs. In neither case are blanks counted.

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    The statute you cite isn't a model of clarity with respect to the issue raised in the question, but I agree that your conclusion is correct. – ohwilleke Nov 2 '18 at 17:39

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